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MECHANICS AND HEAT 



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MECHANICS AND HEAT 



A TEXT BOOK FOR COLLEGES 
AND TECHNICAL SCHOOLS 



bY 

WM. S. FRANKLIN AND BARRY MACNUTT 



/ 



Neto York 
THE MACMILLAN COMPANY 

LONDON: MACMILLAN & CO., LTD. 
1916 

All rights reserved 






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Copyright 1907, 1910 
By THE MACMILLAN COMPANY 



Set up and electrotyped. Published July, 1910 
September, 1916 



Press of 

The New Era Printing Companv 

Lancaster, Pa 



PREFACE. 

The authors have had a very definite point of view in the 
preparation of this elementary treatise on Mechanics and Heat, 
and they assume that those to whom this preface is addressed 
have read their introductory chapter, especially the section en- 
titled The Science of Physics. 

The most important function of the teacher of physics is to 
build the logical and mechanical structure of the science; the 
logical structure mainly by lecture and recitation work including 
a great deal of practice by the student in numerical calculation, 
and the mechanical structure by laboratory work. These two 
phases of physics study should run along together. This text, 
however, is intended as a basis for the work of the class-room. 

In their experience, the authors have come to recognize four 
chief difficulties in the teaching of elementary physics, as follows : 

One difficulty is that the native sense of most men is incapable, 
without stimulation and direction, of supplying the material upon 
which the logical structure of the science is intended to operate. 
It is necessary to direct the student's attention to many familiar 
things and to present in the class-room a great many experimental 
demonstrations of presumably familiar phenomena. 

A second difficulty is that the human mind, intuitively habitu- 
ated as it is to consider the important practical affairs of life, 
can hardly be turned to that minute consideration of apparently 
insignificant details which is so necessary in the scientific handling 
even of the most distinctly practical problems. 

A third difficulty, which indeed runs through the entire front- 
of-progress of the human understanding, is that mind-stuff, which 
has been developed as correlative to certain aspects of our ances- 
tral environment, must be rehabilitated in entirely new relations 
in fitting a man for the conditions of civilized life. Every teacher 
knows how much coercion is required for so little of this rehabili- 



vi PREFACE. 

tation; but the bare possibility of the process is a remarkable 
fact, and that it is possible to the extent of bringing a Newton or 
a Pasteur out of a hunting and fishing ancestry is indeed wonder- 
ful. 

A fourth difficulty is that the possibility of this rehabilitation 
of mind-stuff has grown up as a human faculty almost solely 
on the basis of language, and the essence of this rehabilitation 
is the formation of what we call ideas; whereas a great deal of 
our knowledge of physics is correlated in mechanisms. Familiar- 
ity with machinery is perhaps the most important condition for 
the successful study of the physical sciences, and the mechanisms 
of manipulation and measurement are essential parts of the struc- 
ture of the physical sciences without which the study of physics 
degenerates into an ineffective exercise in meaningless specula- 
tion. 

Everyone is familiar with the life history of the butterfly, 
how it lives first as a caterpillar and then undergoes a complete 
transformation into a winged insect. It is of course evident that 
the bodily organs of the caterpillar are not at all suited to the 
needs of a butterfly, the very food (of those species which take 
food) being entirely different. As a matter of fact, almost every 
portion of the bodily structure of the caterpillar is dissolved as it 
were into a formless pulp at the beginning of the transformation, 
and the organization of a flying insect then grows out from a 
central nucleus very much as a chicken grows in the food-stuff 
of an egg. So it is in the growth of a man. In early childhood, 
if the individual is favored by Fortune, he exercises and develops 
more or less extensively the primitive instincts and modes of the 
race in a free outdoor life, and the experiences so gained are so 
much mind-stuff to be dissolved and transformed under more or 
less coercion and constraint into an effective mind of the twen- 
tieth-century type. 

Many of our teachers, especially those who handle the mathe- 
matical sciences, seem to think that ideas can be built up in 
young men's minds by a sort of hocus pocus, out of nothing; 



PREFACE. vil 

but it seems to the authors that ideas like everything else in this 
world must be made out of something. All elemental knowledge, 
such as the knowing how to throw a ball, how to ride a bicycle, 
how to swim, or how to use a tool, seems to be locked in the 
marginal regions of the mind as a very substantial but very 
highly specialized kind of intuition, and the problem of teaching 
elementary physical science is in part the problem of how, by 
suggestion or otherwise, to drag this material into the field of 
consciousness where it may be transformed into a generalized 
logical structure. A formal and abstract presentation of the 
principles of elementary science tends more than anything else 
to inhibit the influx of this elemental knowledge from the marginal 
regions of the mind into the field of consciousness and results in 
the building up of a theoretical structure which can have no 
traffic with any mental field beyond its own narrow boundaries. 
Such a state of mind is but a kind of idiocy, and to call it a 
knowledge of science is silly scholasticism. 

The best way to meet this quadruply difficult situation is to 
relate the teaching of the physical sciences as much as possible 
to the immediately practical things of life, and to go in for sug- 
gestiveness as the only way to avoid a total inhibition of the 
sense that is born with our students. It is not, however, a ques- 
tion of exactitude versus suggestiveness, for, indeed, both are 
necessary ; exactitude relates chiefly to the realm of ideas whereas 
suggestiveness relates chiefly to the realms of objective reality. 
That is to say, suggestiveness is the One form of appeal to the 
rudimentary and elemental things of the mind which are more 
directly connected with objective reality than are the more highly 
abstracted ideas which operate almost wholly within the field of 
consciousness. 

Such a method is certainly calculated to limber up our theories 
and put tham all at work, the pragmatic* method our friends the 
philosophers call it, a method which pretends to a conquering 

*From the Greek word Trpayfta, meaning action, from which our words practice 
and practical come. 



viii PREFACE. 

destiny.* Whatever one may think of that philosophy of life 
which exhibits itself in a temperament of the most intensely 
practical and matter-to-fact type, it is certain that pragmatism 
is the only philosophy a science teacher can entertain and escape 
what to him is the most dangerous form of idolatry, science for 
its own sake. 

The author's acknowledgements are due to S. LeRoy Brown 
for assistance in reading proof, and to R. L. Charles, C. M. Kilby, 
S. LeRoy Brown, A. C. Callen, and J. L. Dynan for many sug- 
gestions as to the arrangement of the text. 

W. S." Franklin, 
Barry MacNutt. 

South Bethlehem, Pa., 
June 22, 1910. 

* See the interesting little book on Pragmatism, by William James, Henry Holt 
& Co. 



TABLE OF CONTENTS. 

PAGES 

INTRODUCTION 1-15 

PART I. ELEMENTS OF MECHANICS. 

CHAPTER I. 
Weights and Measures 19-25 

CHAPTER II. 
Physical Arithmetic 26-49 

CHAPTER III. 
Simple Statics , 50-74 

CHAPTER IV. 
Dynamics. Translatory Motion. 75~ii7 

CHAPTER V. 
Friction. Work and Energy 1 18-142 

CHAPTER VI. 
Rotatory Motion 143-182 

CHAPTER VII. 
Elasticity (Statics) 183-218 

CHAPTER VIII. 
Hydrostatics 219-238 

CHAPTER IX. 
Hydraulics 239-269 

PART II. THEORY OF HEAT. 

CHAPTER I. 
Temperature. Thermal Expansion 273-303 

CHAPTER II. 

Calorimetry 304-314 

ix 



x TABLE OF CONTENTS. 

CHAPTER III. 
Thermal Properties of Solids, Liquids and Gases 315-342 

CHAPTER IV. 
The Atomic Theory of Gases 343~349 

CHAPTER V. 
The Second Law of Thermodynamics 350-375 

CHAPTER VI. 
Further Developments of Thermodynamics 376-397 

CHAPTER VII. 

Transfer of Heat : 398-403 

Index 404 



ELEMENTS OF MECHANICS. 



INTRODUCTION. 



Everyone knows the capability of the Indian for long-continued 
and serious effort in his primitive mode of life, and yet it is dif- 
ficult to persuade an Indian "farmer" to plow. Everyone knows, 
also, that the typical college student is not stupid, and yet 
it is difficult to persuade the young men of practical and busi- 
ness ideals in our colleges and technical schools to study the 
abstract elements of science. Indeed, it is as difficult to get 
these young men to hold abstract things in mind as it is to get a 
young Indian to plow, and for almost exactly the same reason. 
The scientific handling of any problem requires a minute and 
painstaking consideration of details which are in themselves de- 
void of human value; and this quality of detachment, which 
pervades every branch of science, is the most serious obstacle to 
young people in their study of the sciences. But analysis is 
necessary, and it is the object of this introduction to convey to 
the student some understanding of this fact. 

There is perhaps a narrowing tendency in the necessarily 
specialized courses of study of engineering and professional stu- 
dents, and the correction for this tendency would seem to lie 
outside of science study. However this may be, it is of the 
greatest importance that every student should realize two things 
in connection with his study of the physical sciences. The first 
is that the study of the physical sciences is exacting beyond all 
compromise, involving as it does a degree of coercion and con- 
straint which it is beyond the power of any teacher greatly to 
mitigate; and second is that the completest science stands 
abashed before the infinitely complicated and fluid array of 



2 ELEMENTS OF MECHANICS. 

phenomena of the material world, except only in the assurance 
which its method gives. 

There is a kind of salamander, the axolotl, which lives a tadpole- 
like youth and never changes to the adult form unless a stress of dry 
weather annihilates his watery world ; but he lives always and re- 
produces his kind as a tadpole, and a very odd-looking tadpole 
he is, with his lungs hanging from the sides of his head in feathery 
tassels, brownish in tint from the red blood inside and the mud 
that settles on the outside of the tiny streamers. When the 
aquatic home of the axolotl dries up, he quickly develops a pair 
of internal lungs, lops off his tassels and embarks on a new mode 
of life on land. If our young men are to develop beyond the 
tadpole stage they must meet with quick and responsive inward 
growth, that new and increasing "stress of dryness" as many are 
wont to call our modern age of science and industry. The 
purely impersonal character of science study represents a pro- 
found change in the things a young man is called upon to 
think about and in the character of his thinking, but the 
popular conception of the specialist as a "Mr. Dry-as-dust" is 
a caricature, for the dryness of science is by no means the same 
thing as the uninteresting quality of boredom. 

The Laws of Motion. 

The most prominent aspect of all phenomena is motion. In 
that realm of nature which is not of man's devising* motion is 
universal. In the other realm of nature, the realm of things 
devised, motion is no less prominent. Every purpose of our 
practical life is accomplished by movements of the body and by 
directed movements of tools and mechanisms. 

The Jaws of motion. Every one has a sense of the absurdity 

* Science as young people study it has two chief aspects, or in other words, it 
may be roughly divided into two parts, namely, the study of the things which come 
upon us, as it were, and the study of the things which we deliberately devise. The 
things that come upon us include weather phenomena and every aspect and phase 
of the natural world, the things we cannot escape; and the things we devise relate 
chiefly to the serious work of the world, the things we laboriously build and the 
things we deliberately and patiently seek. 



INTRODUCTION. 3 

of the idea of reducing the more complicated phenomena of 
nature to an orderly system of mechanical law. For to speak 
of motion is to call to mind first of all the phenomena that are 
associated with the excessively complicated, incessantly changing, 
turbulent and tumbling motion of wind and water. These phe- 
nomena have always had the most insistent appeal to us, they 
have confronted us everywhere and always, and life is an un- 
ending contest with their fortuitous diversity. The laws of 
motion! Let us consider the impetuous complexity of a great 
storm or the dreadful confusion of a railway wreck and under- 
stand that what we call the laws of motion, although they have 
a great deal to do with the ways in which we think, have very 
little to do with the phenomena of nature. The laws of motion! 
Truly the science of mechanics is circumscribed in utter repudia- 
tion of such universal phrase, and yet the ideas which constitute 
the laws of motion have an almost unlimited extent of legitimate 
range, and these ideas must be possessed with a perfect precision 
if one is to acquire any solid knowledge whatever of the phenomena 
of motion. The necessity of precise ideas. Herein lies the im- 
possibility of compromise and the necessity of coercion and con- 
straint; one must think so and so, there is no other way. Indeed 
there is always a conflict in the mind of even the most willing 
student due to the constraint which precise ideas place upon our 
vivid and primitively adequate sense of physical things. This 
conflict is perennial and it is by no means a one-sided conflict 
between mere crudity and refinement, for refinement ignores 
many things. Indeed, precise ideas not only help to form* our 
sense of the world in which we live but they inhibit sense as well 
and their rigid and unchallenged rule would be indeed a stress of 
dryness. 

The laws of motion. We return again and yet again to the 
subject, for one is not to be deterred therefrom by any concession 
of inadequacy, no, nor by any degree of respect for the vivid 
youthful sense of those things which to suit our narrow purpose 

* See discussion of Bacon's New Engine on page 14. 



4 ELEMENTS OF MECHANICS. 

must be stripped completely bare. It is unfortunate, however, 
that the most familiar type of motion, the flowing of water and 
the blowing of the wind, is bewilderingly useless as a basis for 
the establishment of the simple and precise ideas which are 
called the "laws of motion," and which are the most important 
of the fundamental principles of physics. These ideas have in 
fact grown out of the study of the simple phenomena which are 
associated with the motion of bodies in bulk without perceptible 
change of form, the motion of rigid bodies, so-called. 

Before narrowing down the scope of the discussion, however, 
let us illustrate a very general application of the simplest idea 
of motion, the idea of velocity. Every one has, no doubt, an 
idea of what is meant by the velocity of the wind; and a sailor, 
having what he calls a ten-knot wind, knows that he can manage 
his boat with a certain spread of canvas and that he can accom- 
plish a certain portion of his voyage in a given time; but an 
experienced sailor, although he speaks glibly of a ten-knot wind, 
belies his speech by taking wise precaution against every con- 
ceivable emergency. He knows that a ten-knot wind is by no 
means a sure or a simple thing with its incessant blasts and whirls ; 
and a sensitive anemometer, having more regard for minutiae 
than any sailor, usually registers in every wind a number of 
almost complete but excessively irregular stops and starts every 
minute and variations of direction that sweep around half the 
horizon ! 

We must evidently direct our attention to something simpler 
than the wind. Let us, therefore, consider the drawing of a 
wagon or the propulsion of a boat. It is a familiar experience 
that effort is required to start a body moving and that continued 
effort is required to maintain the motion. Certain very simple 
facts as to the nature of this effort, as to the amount of effort 
required to produce motion and as to the conditions which deter- 
mine the amount of effort required to keep a body in motion, 
were discovered by Sir Isaac Newton, and on the basis of these 
facts Newton formulated the "laws of motion." 



INTRODUCTION. 5 

The effort required to start a body or to keep it moving is 
called force. Thus, if one starts a box sliding along a table one 
is said to exert a force on the box. The same effect might be 
accomplished by interposing a stick between the hand and the 
box, in which case one would exert a force on the stick and the 
stick in its turn would exert a force on the box. We thus arrive at 
the notion of force action between inanimate bodies, between the 
stick and the box in this case, and Newton pointed out that the 
force action between the two bodies A and B always consists 
of two equal and opposite forces, that is to say, if body A exerts 
a force on B, then B exerts an equal and opposite force on A, 
or, to use Newton's words, action is equal to reaction and in a 
contrary direction. 

In leading up to this statement one might consider the force 
with which a person pushes on the box and the equal and op- 
posite force with which the box pushes back on the person, but 
if one does not wish to introduce the stick as an intermediary, 
it is better to speak of the force with which the hand pushes on 
the box, and the equal and opposite force with which the box 
pushes back on the hand, because in discussing physical phe- 
nomena it is of the utmost importance to pay attention to im- 
personal things. Indeed our modern industrial life, in bringing 
men face to face with an entirely unprecedented array of intricate 
mechanical and physical problems, demands of every one a great 
and increasing amount of impersonal thinking, and the precise 
and rigorous modes of thought of the physical sciences are being 
forced upon widening circles of men with a relentless insistance 
— all of which it was intended to imply by referring to the stress 
of dryness which overtakes the little axolotl in his contented 
existence as a tadpole. 

When we examine into the conditions under which a body 
starts to move and the conditions under which a body once 
started is kept in motion, we come across a very remarkable 
fact, if we are careful to consider every force which acts upon the 
body, and this remarkable fact is that the forces which act upon a 



6 ELEMENTS OF MECHANICS. 

body which remains at rest are related to each other in precisely 
the same way as the forces which act upon a body which con- 
tinues to move steadily along a straight path. Therefore, it is 
convenient to consider, first the relation between the forces which 
act upon a body at rest, or upon a body in uniform motion, and 
then to consider the relation between the forces which act upon 
a body which is starting or stopping or changing the direction 
of its motion. 

Suppose a person A were to hold a box in mid-air. To do so it 
would of course be necessary for him to push up on the box so 
as to balance the downward pull of the earth, the weight of the 
box, as it is called. Then if another person B were to take hold 
of the box and pull upon it in any direction, A would have to 
exert an equal pull on the box in the opposite direction to keep 
it stationary. That is to say, the forces which act upon a sta- 
tionary body are always balanced. 

Every one, perhaps, realizes that what is here said about the 
balanced relation of the forces which act upon a stationary box, 
is equally true of the forces which act on a box similarly held in 
a steadily moving railway car or boat. Therefore, the forces 
which act upon a body which moves steadily along a straight 
path are balanced. 

This is evidently true when the moving body is surrounded 
on all sides by things which are moving along with it, as in a car 
or a boat; but how about a body which moves steadily in a 
straight path but which is surrounded by bodies which do not 
move along with it? Everyone knows that some active agent 
such as a horse or a steam engine must pull steadily upon such 
a body to keep it in motion, and that if left to itself such a moving 
body quickly comes to rest. Many have, no doubt, reached 
this further inference from experience, namely, that the tendency 
of moving bodies to come to rest is due to the dragging forces, 
or friction, with which surrounding bodies act upon a body in 
motion. Thus a moving boat is brought to rest by the drag of 
the water when the propelling force ceases to act; a train of cars 



INTRODUCTION. 7 

is brought to rest because of the drag due to friction when the 
pull of the locomotive ceases; a box which is moved across a 
table comes to rest when left to itself, because of the drag due 
to friction between the box and the table. 

We must, therefore, always consider two distinct forces when 
we are concerned with a body which is kept in motion, namely, 
the propelling force due to some active agent such as a horse or 
an engine, and the dragging force due to surrounding bodies. 
Newton pointed out that when a body is moving steadily along 
a straight path, the propelling force is always equal and opposite 
to the dragging force. Therefore, The forces which act upon 
a body which is stationary, or which is moving uniformly along 
a straight path, are balanced forces. 

Many hesitate to accept as a fact the complete and exact 
balance of propelling and dragging forces on a body which is 
moving steadily along a straight path in the open, but direct 
experiment shows it to be true, and the most elaborate calcula- 
tions and inferences based upon this notion of the complete 
balance of propelling and dragging forces on a body in uniform 
motion are verified by experiment. One may ask, why a canal 
boat, for example, should continue to move if the pull of the mule 
does not exceed the drag of the water; but why should it stop 
if the drag does not exceed the pull? Understand that we are 
not considering the starting of the boat. The fact is that the 
conscious effort which one must exert to drive a mule, the cost 
of the mule, and the expense of his keep, are what most people 
think of, however hard one tries to direct their attention solely 
to the state of tension in the rope that hitches the mule to the 
boat after the boat is in full motion; and most people consider 
that if the function of the mule is simply to balance the drag of 
the water so as to keep the boat from stopping, then why should 
there not be some way to avoid the cost of so insignificant an 
operation? There is, indeed, an extremely important matter in- 
volved here which we will consider when we come to the discussion 
of work and energy; but it has no bearing on the matter of the 



8 ELEMENTS OF MECHANICS. 

balance of propulsion and drag on a body which moves steadily 
along a straight path. 

Let us now consider the relation between the forces which act 
upon a body which is changing its speed, upon a body which is 
being started or stopped, for example. Everyone has noticed 
how a mule strains at his rope when starting a canal boat, 
especially if the boat is heavily loaded, and how the boat con- 
tinues to move for a long time after the mule ceases to pull. In 
the first case, the pull of the mule greatly exceeds the drag of the 
water, and the speed of the boat increases; and in the second 
case, the drag of the water of course exceeds the pull of the mule, 
for the mule is not pulling at all, and the speed of the boat de- 
creases. When the speed of a body is changing, the forces which 
act on the body are unbalanced, and we may conclude that the 
effect of an unbalanced force acting on a body is to change the 
velocity of the body; and it is evident that the longer the unbal- 
anced force continues to act the greater the change of velocity. 
Thus if the mule ceases to pull on a canal boat for one second the 
velocity of the boat will be but slightly reduced by the unbalanced 
drag of the water, whereas if the mule ceases to pull for two 
seconds the decrease of velocity will be much greater. In fact 
the change of velocity due to a given unbalanced force is proportional 
to the time that the force continues to act. This is exemplified by a 
body falling under the action of the unbalanced pull of the earth ; 
after one second it will have gained a certain amount of velocity 
(about 32 feet per second), after two seconds it will have made a 
total gain of twice as much velocity (about 64 feet per second), 
and so on. Furthermore, since the velocity produced by an un- 
balanced force is proportional to the time that the force continues 
to act, it is evident that the effect of the force should be specified 
as so-much-velocity-produced-per-second, exactly as in the case of 
earning money, the amount one earns is proportional to the length 
of time that one continues to work, and we always specify one's 
earning capacity as so-much-money-earned-per-day.* 

* See footnote on page 37. 



INTRODUCTION. 9 

Everyone knows what it means to give an easy pull or a hard 
pull on a body. That is to say, we all have the ideas of greater 
and less as applied to forces. Everybody knows also that if a 
mule pulls hard on a canal boat, the boat will get under way more 
quickly than if the pull is easy, that is, the boat will gain more 
velocity per unit of time under the action of a hard pull than 
under the action of an easy pull. Therefore, any precise state- 
ment of the effect of an unbalanced force on a given body must 
correlate the precise value of the force and the exact amount of 
velocity produced per unit of time by the force. This seems a 
very difficult thing, but its apparent difficulty is very largely due 
to the fact that as yet we have not agreed as to what we are to 
understand by the statement that one force is precisely three, or 
four, or any number of times as great as another. Suppose, there- 
fore, that we agree to call one force twice as large as another when 
it will produce in a given body twice as much velocity in a given 
time (remembering of course that we are now talking about un- 
balanced forces, or that we are assuming for the sake of simplicity 
of statement, that no dragging forces exist). As a result of this 
definition we may state that the amount of velocity produced 
per second in a given body by an unbalanced force is proportional 
to the force. 

Of course we know no more about the matter in hand than we 
did before we adopted the definition, but we do have a good illus- 
tration of how important a part is played in the study of science, 
by what we may call making up one's mind, in the sense of put- 
ting one's mind in order. This kind of thing is very prominent 
in the study of elementary physics, and that rather indefinite 
reference, in the story of the little tasseled tadpole, to an inward 
growth so needful before one can hope for any measure of success 
in our modern world of scientific industry was an allusion to 
this thing, the "making-up" of one's mind. Nothing is so es- 
sential in the acquirement of exact and solid knowledge as the 
possession of precise ideas, not indeed that a perfect precision is 
necessary as a means for retaining knowledge, but that nothing 



IO ELEMENTS OF MECHANICS. 

else so effectually opens the mind for the perception even of the 
simplest evidences of a subject.* 

We have now settled the question as to the effect of different 
unbalanced forces on a given body on the basis of a very general 
experience, and by an agreement as to the precise meaning to be 
attached to the statement that one force is so many times as 
great as another ; but how about the effect of the same force upon 
different bodies, and how may we identify the force so as to be 
sure that it is the same? It is required, for example, to exert a 
given force on body A and then to exert the same force on another 
body B. This can be done by causing a third body C (a coiled 
spring, for example) to exert the force; then the forces exerted 
on A and B are the same if the reaction in each case produces the 
same effect on body C (the same degree of stretch, for example). 
Concerning the effects of the same unbalanced force on different 
bodies three things have to be settled by experiment as follows: 

(a) In the first place let us suppose that a certain force F is 
twice as large as a certain other force G, according to our agree- 
ment, because the force F produces twice as much velocity every 
second as force G when the one and then the other of these forces 
is caused to act upon a given body, a piece of lead for example. 
Then, does the force F produce twice as much velocity every 
second as the force G whatever the nature and size of the given 
body, whether it be wood, or ice, or sugar? Experiment shows 
that it does. 

(&) In the second place, suppose that we have such amounts 
of lead, of iron, of wood, etc., that a certain given force produces 
the same amount of velocity per second when it is made to act, 
as an unbalanced force, upon one or another of these various 
bodies. Then what is the relation between the amounts of these 
various substances? Experiment shows that they all have the 
same mass in grams, or pounds, as determined by a balance. 

* Opens the mind, that is, for those things which are conformable to or consistent 
with the ideas. The history of science presents many cases where accepted ideas 
have closed the mind to contrary evidences for many generations. Let young men 
beware ! 



INTRODUCTION. 1 1 

That is, a given force produces the same amount of velocity per 
second in a given number of grams of any kind of substance. 
Thus the earth pulls with a certain definite force (in a given 
locality) upon M grams of any substance and, aside from the 
dragging forces due to air friction, all kinds of bodies gain the 
same amount of velocity per second when they fall under action 
of the unbalanced pull of the earth. 

(c) In the third place, what is the relation between the velocity 
per second produced by a given force and the mass in grams (or 
pounds) of the body upon which it acts? Experiment shows that 
the velocity per second produced by a given force is inversely 
proportional to the mass of the body upon which the force acts. 
In speaking of the mass of a body in grams (or pounds) we here 
refer to the result which is obtained by weighing the body on a 
balance scale, and the experimental fact which is here referred to 
constitutes a very important discovery: namely, when one body 
has twice the mass of another, according to the balance method 
of measuring mass, it is accelerated half as fast by a given un- 
balanced force. 

The effect of an unbalanced force in producing velocity may 
therefore be summed up as follows: The velocity per second 
produced by an unbalanced force is proportional to the force 
and inversely proportional to the mass of the body upon which 
the force acts. Furthermore, the velocity produced by an un- 
balanced force is always in the direction of the force. 

Physical Measurement. 

Among primitive races all things subject to exchange or bar- 
ter are estimated by simple counting. Thus a Tartar herds- 
man estimates his wealth by counting his cattle. With the 
growth of civilization, however, there has been a great increase 
in the variety of useful and exchangeable commodities, and many 
of these commodities cannot be estimated by simple counting. 
The result has been that the simple operation of counting, which, 
of course, can be applied only to groups of separate and distinct 



12 ELEMENTS OF MECHANICS. 

things, has developed into the operation called measurement, in 
which a continuous whole is estimated numerically by dividing 
it into equal, unit parts, and counting these parts. Thus oil 
or wine is counted out by means of a gallon measure, and cloth 
is counted out by means of a yard-stick. 

In many kinds of measurement, the two distinct operations, 
(a) dividing into equal unit parts and (b) counting are obscured by 
the use of more or less elaborate measuring devices, but every 
measurement of whatever kind does, in fact, consist of these two 
fundamental operations. Thus in measuring a length by means 
of a scale of inches, the operation of dividing into unit parts has 
been performed once for all by the maker of the scale, and in 
this case the operation of counting is, in large part, "ready-made" 
by the numbers stamped on the scale. In the weighing of a con- 
signment of coal, the operation of dividing into unit parts has 
been performed once for all by the maker of the set of weights 
and of the divided balance beam, and the operation of counting 
is, in large part, "ready-made" by the numbers stamped upon 
the weights and upon the beam. 

The long experience of the race in estimating by the simple 
counting of separate things has given rise to a sense of sharp dis- 
tinction between any two numbers, thus iooo horses is clearly 
not the same thing as 999 horses; but this sharp distinction be- 
tween approximately equal numbers is devoid of physical signifi- 
cance in the case of numbers derived by measurement, because of 
the approximate character of the operation of dividing a whole 
into unit parts. A person might buy a herd of horses supposing 
the number to be 1000, whereas a correct count would show 999; 
and although the purchaser might reasonably say, "Oh, let it go; 
it makes no difference," still the fact would remain that 999 
horses is not 1000 horses; suppose, however, a man were to buy 
1000 yards of cloth, he might re measure the cloth and count 
999 yards, but in remeasuring, the day may have been damp, or 
he may not have stretched the cloth in the same way as the man- 
ufacturer, or he may have taken more or less pains in fitting the 



INTRODUCTION. 1 3 

yard-stick to the successive portions of the cloth, or his yard- 
stick may have been in error. It is in fact impossible* to show 
that iooo yards of cloth is not 999 yards of cloth, except by 
reasoning that 1000 pieces of silver is not 999 pieces of silver. 
A yard of cloth is not a separate thing whereas a piece of silver is. 

The operation of dividing a length or an angle into equal unit 
parts for the purpose of measurement is an operation of fitting a 
standard to each part, an operation of congruence; and the 
actual measurement of any physical whole — let us not speak of 
it as a quantity until we have attached a number to it — depends 
upon one or another variety of congruence as a basis for the 
assumption of equality of the parts which are to be counted. 
Thus a pendulum may be assumed to mark off equal intervals 
of time because each movement of the pendulum is like the one 
that follows ; and the equal arm balance is a device for indicating 
a certain kind of congruence between the body which is being 
weighed and the combination of weights which balances it. 

The fundamental meaning of a physical quantity originates in 
and is defined by the actual operation of measuring that quantity. 
Thus, the mass of a body, as a quantity, is defined by the opera- 
tion of weighing by a balance; and, since the result of this 
operation is always the same, within the limits of error, for a 
given amount of any substance, it- is permissible to use this 
result as a measure of the amount of the substance. Nearly 
every physical definition, rightly understood, is an actual physical 
operation. 

The Science of Physics. 

"We advise all men" says Bacon "to think of the true ends 
of knowledge, and that they endeavor not after it for curiosity, 
contention, or the sake of despising others, nor yet for reputation 

*The principles involved in the measurement of cloth do not differ in any respect 
from the principles involved in more precise measurements in the laboratory. 
Every student of physics should be to some extent familiar with the theory of 
errors of observation. Among the best books on this subject are Holman's Pre- 
cision of Measurements (Wiley & Sons) and Merriman's Least Squares (Wiley & 
Sons). 



14 ELEMENTS OF MECHANICS. 

or power or any other such inferior consideration, but solely for 
the occasions and uses of life." It is difficult to imagine any 
other basis upon which the study of physics can be justified than 
for the occasions and uses ol life; in a certain broad sense, indeed, 
there is no other justification. But the great majority of men 
must needs be practical in the narrow sense, and physics, as they 
s^udy it, relates chiefly to the conditions which have been elabo- 
rated through the devices of industry as exemplified in our mills 
and factories, in our machinery of transportation, in optical and 
musical instruments, in the means for the supply of power, heat, 
light, and water for general and domestic use, arid so on. 

From this narrow practical point of view it may seem that 
there can be nothing very exacting in the study of the physical 
sciences; but was it physics? That is the question. One defi- 
nition at least is to be repudiated; it is not "The science of 
masses, molecules and the ether." Bodies have mass and rail- 
ways have length, and to speak of physics as the science of masses 
is as silly as to define railroading as the practice of lengths, and 
nothing as reasonable as this can be said in favor of the concep- 
tion of physics as the science of molecules and the ether; it is the 
sickliest possible notion of physics, whereas the healthiest notion, 
even if a student does not wholly grasp it, is that physics is the 
science of the ways of taking hold of things and pushing them ! 

Bacon long ago listed in his quaint way the things which seemed 
to him most needful for the advancement of learning. Among 
other things he mentioned U A New Engine or a Help to the 
mind corresponding to Tools for the hand,'' 1 and the most remark- 
able aspect of physical science is that aspect in which it consti- 
tutes a realization of this New Engine of Bacon. We continually 
force upon the extremely meager data obtained directly through 
our senses, an interpretation which, in its complexity and pene- 
tration, would seem to be entirely incommensurate with the data 
themselves, and we exercise over physical things a kind of rational 
control which greatly transcends the native cunning of the hand. 
The possibility of .this forced interpretation and of this rational 



INTRODUCTION. 1 5 

control depends upon the use of two complexes: (a) A logical 
structure, that is to say, a body of mathematical and conceptual 
theory which is brought to bear upon the immediate materials of 
sense, and (b) a mechanical structure, that is to say, either (i) a 
carefully planned arrangement of apparatus, such as is always 
necessary in making physical measurements, or (2) a carefully 
planned order of operations, such as the successive operations of 
solution, reaction, precipitation, nitration, and weighing in 
chemistry. 

These two complexes do indeed constitute a New Engine which 
helps the mind as tools help the hand, it is through the enrich- 
ment of the materials of sense by the operation of this New 
Engine that the elaborate interpretations of the physical sciences 
are made possible, and the study of elementary physics is intended 
to lead to the realization of this New Engine : (a) By the building 
up in the mind, of the logical structure of the physical sciences; 
(b) by training in the making of measurements and in the per- 
formance of ordered operations, and (c) by exercises in the appli- 
cation of these things to the actual phenomena of physics and 
chemistry at every step and all of the time with every possible 
variation. 

That, surely, is a sufficiently exacting program; and the only 
alternative is to place the student under the instruction of Jules 
Verne where he need not trouble himself about foundations but 
may follow his teacher pleasantly on a care-free trip to the moon 
or with easy improvidence embark on a voyage of twenty-thou- 
sand leagues under the sea. 



PART I. 

MECHANICS. 



"Our method is continually to dwell among things soberly, 
without abstracting or setting the mind farther from them than 
makes their images meet," and "the capital precept for the whole 
undertaking is that the eye of the mind be never taken off from 
things themselves, but receive their images as they truly are, and 
God forbid that we should offer the dreams of fancy for a model 
of the world." — Bacon. 



CHAPTER I. 

WEIGHTS AND MEASURES. 

1. Length. The meter is the length, at the temperature of 
melting ice, of a certain platinum bar which is preserved in the 
vaults of the International Bureau of Weights and Measures 
near Paris.* A very accurate copy f of this bar is deposited in 
the United States Bureau of Standards in Washington and this 
copy is the legal meter in the United States. 

The yard is defined legally as 3600/3937 of a meter. J 

English Units of Length Metric Units of Length 

1 yard = 3 feet 1 kilometer =1000 meters 

1 foot =12 inches i meter =100 centimeters 

1 mile =1760 yards or 5280 feet 1 meter =1000 millimeters 

1 inch =2.54 centimeters 

1 centimeter =0.3937 inch. 

2. Angle. The angle all the way round a point, that is the 
angle which is represented by the entire circumference of a circle, 
is called a perigon. The perigon is a natural unit of angle and it 
is not necessary to preserve a material copy of it. The unit of 
angle which is universally used on divided circles is the degree; 
it is equal to 1/360 of a perigon. 

*It was intended originally that the meter should be equal to one-ten-millionth 
part of the distance from the equator to either pole of the earth, but the extreme 
difficulty of reproducing accurate copies of the meter on the basis of this definition 
makes the definition not only impracticable but illusory. 

t In fact there are three so-called prototype meter-bars in Washington. For a de- 
scription of the International prototypes of the meter and of the kilogram, see 
Nature, Vol. 51, page 420, February 28, 1895. 

tOn April 5, 1893, a decision was reached by the United States .Superintendent 
of Weights and Measures, with the approval of the Secretary of the Treasury, that 
the meter and the kilogram would be regarded as the fundamental standards not 
only for metric units but also for the customary units of length and mass. See a 
History of the Standard Weights and Measures of the United States by Louis A. 
Fischer, Vol. I., pp. 365-381, Bulletin of the Bureau of Standards (United States 
Department of Commerce and Labor). 

19 



20 ELEMENTS OF MECHANICS. 

In many calculations it is convenient to express angle as follows : 
Imagine a circle of radius r drawn with its center at the apex of 
an angle, and let a be the length of the arc of the circle which is 
included between the boundaries of the angle; then the ratio 
a/r has a fixed value for a given angle, and the value of this 
ratio is frequently used as a numerical measure of the angle. 
The unit angle in this system is the angle of which the length of 
the subtending arc is equal to the radius and it is called the radian. 

One perigon =27r radians = 360 degrees. 

3. Area. The unit of area most extensively used is the area 
of a square of which the side is of unit length.* 

English Units of Area Metric Units of Area 

Areas are usually expressed in square Areas are usually expressed i n square 
inches, or square feet, or square centimeters or in square meters, 

yards. 

1 square inch =6.45 square centimeters. 

1 square centimeter =0.155 square inches. 

4. Volume or Capacity. The unit of volume most extensively 
used is the volume of a cube of which the edge is of unit length. 

English Units of Volume Metric Units of Volume 

Volumes are frequently expressed, in Volumes are frequently expressed in 
cubic inches, or in cubic feet or cubic centimeters, or in cubic 

in cubic yards. meters. 

1 gallon =4 quarts = 23 1 cubic inches 1 liter =1000 cubic centimeters 

1 quart =1.36 liter 

1 liter =0.88 quarts 

1 cubic inch =16.38 cubic centimeters 
1 cubic centimeter =0.061 cubic inch 

5. Mass. Every one is familiar with the measurement of 
materials by volume and by weight, but every one does not dis- 
tinguish between the two methods in common use for measuring 
"by weight," namely, (a) the method in which the spring scale is 
used, and (b) the method in which the balance scale is used. The 
spring scale measures the force with which the earth pulls a body ; 
but the force with which the earth pulls a given body is slightly 

*The circular mil, much used by electricians as a unit area, is the area of a circle 
one mil (1/1000 inch) in diameter. The area of any circle in circular mils is equal 
to the square of its diameter in mils. 



WEIGHTS AND MEASURES. 21 

different at different places on the earth, and therefore the weight 
of a given body, as indicated by a spring scale, varies with location. 
The indication of the balance scale, on the other hand, does not 
vary with the gravity-pull of the earth because the gravity-pull 
on the weights and the gravity-pull on the weighed body both 
change together. The indication of a balance scale is, therefore, 
independent of the value of gravity. The result of the operation 
of weighing a body by a balance scale is called the mass of the body* 

Considering that weighing is nearly always done by the balance 
scale, it is evident that what is popularly called the weight of a 
body is what scientific men call the mass of the body, and it is 
important to remember that the force with which the earth pulls 
a body is called the weight of the body by scientific men. The 
verb to weigh nearly always means to determine the mass of a 
body by means of a balance scale. 

Units of Mass. The kilogram is the mass of a certain piece 
of platinum which is deposited in Paris. A very accurate copy f 
of this kilogram is deposited in the United' States Bureau of 
Standards J in Washington and this copy is the legal kilogram in 
the United States. 

The pound avoirdupois is defined legally as 1/2.204622 of a 
kilogram. 

English Units of Mass. Metric Units of Mass. 

1 ton =2000 pounds I metric-ton =1000 kilograms 

1 pound =16 ounces 1 kilogram =1000 grams 

1 pound = 7000 grains 1 gram =1000 milligrams 

*The mass of a body is sometimes denned on the basis of Newton's second law 
of motion. Thus a body may be said to have a mass of 90 pounds when a given 
unbalanced force acting on the body produces 1 /90 as much acceleration as the same 
force would produce when acting as an unbalanced force upon a one-pound body. 
In all practical work, however, and in nearly all scientific work the mass of a body 
is determined by a balance scale and therefore the above definition of mass is the 
fundamental one because it refers to the method which is almost universally used 
for measuring mass. 

fin fact there are three so-called prototype kilograms in Washington. 

JThe Bureau of Standards is organized under the United States Department of 
Commerce and Labor, and the function of the Bureau is to verify weights and 
measures for commerical and scientific use and to carry out researches in various 
branches of physics and chemistry. 



22 ELEMENTS OF MECHANICS. 

i kilogram = 2.205 pounds 

1 pound =0.4536 kilogram. 
62/^ pounds of water = 1 cubic foot nearly. 
1 gram of water = 1 cubic centimeter very nearly. 

Measurement of mass. The analytical balance consists of a 
delicately mounted equal-arm lever with pans suspended from its 
ends. The balance is used simply for indicating the equality of the 
masses of two bodies, that is, two bodies are said to have equal 
masses when they balance each other when suspended from the 
ends of an equal-arm lever. 

The determination of the mass of a body by means of the bal- 
ance depends upon the use of a set of weights which may be com- 
bined in such a way as to match the mass of the body. Such a 
set of weights may be made by taking two pieces of metal weigh- 
ing together one kilogram and then making them balance each 
other by cutting metal off from one and adding the shavings to 
the other, thus giving a half-kilogram weight. Then a quarter- 
kilogram weight may be made in the same way and so on. A set 
of weights more convenient in use is a set which contains a five, a 
two, and two ones of each — units, tens, hundreds, etc., of grams. 

6. Force. The effort exerted to start a body moving or the 
effort which must be continually exerted to keep a body in motion 
when it is opposed by friction is called force. 

One pound of force is the force with which the earth pulls on 
a one-pound body. Thus, to say that a powerful locomotive 
exerts a force of 20,000 pounds in starting a train means that the 
pull of the locomotive is 20,000 times as great as the pull of the 
earth on a one-pound body. 

One kilogram of force is the force with which the earth pulls on 
a one-kilogram body. 

It is evident that the word pound (or kilogram) has a very 
different meaning in the following two cases, (a) when we speak 
of a pound of sugar or 1000 pounds of coal, and (b) when we speak 
of a force of one pound or a force of 1000 pounds. One pound 
of sugar is a perfectly definite amount of sugar, but the pull of 



WEIGHTS AND MEASURES. 23 

the earth on a one-pound body is not the same everywhere so 
that the pound of force is not a perfectly definite* amount of force. 

7. Density and specific gravity. Every one knows that lead 
is heavier than cork and yet everyone feels instantly that the 
question ''Which is heavier, a pound of lead or a pound of cork?" 
is ambiguous. The word heavy has, in fact, two meanings. One 
pound of cork is heavier than one-half pound of lead in the same 
sense that a ton of coal is heavier than a pound of coal. In this 
case the word heavy refers to the amount of material expressed 
in pounds or kilograms. On the other hand lead is heavier than 
cork in the sense that a piece of lead weighs more than an 
equal bulk of cork. In this case the word heavy refers to an 
inherent property of a substance ; the word density is used to desig- 
nate this inherent property of heaviness. Thus lead has a greater 
density than cork. The density of a substance may be specified 
by giving its mass per unit volume. Thus the density of water 
is about 62^/2 pounds per cubic foot, the density of copper is 555 
pounds per cubic foot, and the density of ordinary kerosene is 
about 7 pounds per gallon. In scientific work it is usual to 
specify the density of a substance in grams per cubic centimeter. 

The specific gravity of a substance at a given temperature is the 
ratio of the density of the substance to the density of water at 
the same temperature. Thus if a substance is 2.5 times as heavy 
as an equal bulk of water, the specific gravity of the substance is 
said to be 2.5. 

8. Time. The length of a day (noon to noon) is slightly dif- 
ferent at different seasons of the year, but the average length of a 
day from year to year is very nearly invariable. The second is 
1 /86400th of the average day (noon to noon).f 

*The dyne and the poundal are perfectly definite units of force. This matter is 
discussed at some length in Art. 33. It has been suggested that the weight of a 
pound at a specified place on the earth be adopted as the definite unit of force, 
and that a force in pounds measured at any other place on the earth be multiplied 
by the ratio g'\g where g is the acceleration of gravity at the standard place and 
g' is the acceleration of gravity at the given place. 

fFor a full discussion of the measurement of time see Young's General Astron. 
omy, page 15, Ginn & Co. 



24 ELEMENTS OF MECHANICS. 

Measurement of time. Any movement of a body which repeats 
itself in equal intervals of time is called periodic motion; single 
movements are called vibrations. A vibrating pendulum is the 
most familiar example of periodic motion. If one counts the 
number N of vibrations of a pendulum in a day and the number 
n during a given interval of time, then the interval is equal to 
n/N of a day. 

The clock* is a machine for maintaining the vibrations of a pen- 
dulum and for keeping count of the vibrations. In the simplest 
case the pendulum is adjusted to make 86,400 vibrations in a 
day (one vibration each second), and the clock hands count these 
vibrations in groups of sixty and sixty-times-sixty instead of count- 
ing in tens, hundreds and thousands. 

In portable clocks a balance wheel takes the place of a pendulum. 

Problems. 

1. Reduce an angle of 20 to radians. Reduce an angle of 
one radian to degrees. Ans. 20 equals 0.349 radian; one radian 
equals 57°.2958. 

2. (a) Find the area in circular mills of a wire of which the 
diameter is 0.1 inch, (b) Find the area in circular mils of a 
copper bar yi of an inch thick and ^2 inch wide. Ans. (a) 10,000 
circular mils; (b) 79580 circular mils. 

Note. The area of a rectangle in square mils is found by multiplying together 
the length and breadth of the rectangle in mils, and the result may be reduced to 
circular mils by multiplying by 4/1". 

3. The density of alcohol is 6.35 pounds per gallon. What is 
its density in pounds per cubic inch? What is its density in 
grams per cubic centimeter? Ans. 0.0274 pound per cubic inch; 
0.76 gram per cubic centimeter. 

4. The specific gravity of iron is 7.8. What is its density in 
pounds per cubic inch? What is the mass, in pounds, of an iron 

*For discussion of clock errors and description of clock escapements see Encyclo- 
pedia Britannica, 9th Edition, article Clock. For description of compensated 
chronometer balance wheel and chronometer escapement see Lockyer's book 
entitled Stargazing, pages 175 to 210. 



WEIGHTS AND MEASURES. 2$ 

bar 30 feet long and 5*^ square inches sectional area? Ans. 
0.282 pound per cubic inch; the bar has a mass of 558 pounds. 

Note. The foundryman's rule for finding the mass in pounds of an iron casting 
is to divide the volume of the casting in cubic inches by 4. This is equivalent to 
taking the density of cast iron as X of a pound per cubic inch. 

5. A bottle weighs 50.62 grams empty and 288.93 grams when 
full of water at 21 ° C. What is the cubic contents of the bottle? 
Neglect the buoyant force of the air. Ans. 238.77 cubic centi- 
meters. 

Note. The density cf water at 21 C. is 0.99806 gram per cubic centimeter. 

6. The bottle which is specified in problem 5 weighs 239.2 
grams when filled with oil at 21 ° C. What is the density of the 
oil? Neglect the buoyant force of the air. Ans. 0.79 gram per 
cubic centimeter. 



CHAPTER II. 

PHYSICAL ARITHMETIC. 

9. Measures; units. In the expression of a physical quantity 
two factors always occur, a numerical factor and a unit. The 
numerical factor is called the measure of the quantity. Thus a 
certain length is 65 centimeters, a certain time interval is 250 
seconds, a certain electric current is 25 amperes', a certain elec- 
tromotive force is no volts. 

It is a great help towards a clear understanding of physical 
calculations to consider that both units and measures are involved 
in a product of two physical quantities or in a quotient of two 
physical quantities. Thus a rectangle is 5 centimeters wide and 
10 centimeters long; and its area is 5 centimeters times 10 centi- 
meters, which is equal to 50 square centimeters. A cylinder is 
10 centimeters long and the area of one of its ends is 25 square 
centimeters; and its volume is 10 centimeters times 25 square 
centimeters, which is equal to 250 cubic centimeters. A train 
travels 500 feet in 10 seconds and its average velocity during the 
time is 500 feet divided by 10 seconds which is equal to 50 feet 
per second. A body is dragged through a distance of 15 feet by 
a force of 10 pounds and the amount of work done is 15 feet 
times 10 pounds, which is equal to 150 foot-pounds. The word 
per connecting the names of two units indicates that the unit fol- 
lowing is a divisor, thus a velocity of 50 feet per second may be, 
and often is written 50 feet /second. A hyphen connecting the 
names of two units indicates a product of the units ; products and 
quotients of units arrived at in this way are always new physical 
units. Thus the foot per second is a unit of velocity, the foot- 
pound is a unit of work. 

It is important to carry the units through with every numerical 
calculation, the arithmetical operations among the various units 

26 



PHYSICAL ARITHMETIC. 2J 

being indicated algebraically. When this is done there can be no 
ambiguity as to the meaning of the result, and when this is not 
done the result has, strictly speaking, no physical meaning at all. 
Although the unit in terms of which a result is expressed is 
known* when the units are carried through a numerical calcula- 
tion, it frequently happens that the unit is so entirely novel that 
it might almost as well be unknown. Thus the rule for finding 
the area of a rectangle by taking the product of length and 
breadth is entirely general, no matter what units of length are 
used, and the area of a rectangle 2 meters long and 50 centimeters 
wide is equal to 2 meters X 50 centimeters or 100 meter-centi- 
meters. Now the meter-centimeter is a unit of area equal to the 
area of a rectangle one meter long and one centimeter wide and it 
is so entirely unfamiliar as a unit of area among men engaged in 
practical work that one might almost as well not know the value 
of an area at all as to have it given in terms of such a unit. It is, 
for this reason, nearly always necessary to reduce the data of a 
problem to certain accepted units before these data can be 
used intelligibly in numerical calculations. 

10. Units, fundamental and derived. The fundamental physi- 
cal units are those which are fixed by arbitrary preserved stand- 
ards. Thus the unit of length is preserved as a platinum bar in 
Paris, the unit of mass is preserved by a piece of platinum in 
Paris, and the second is naturally preserved in the constancy of 
speed of rotation of the earth, f 

Derived physical units are those which are defined in terms of 
the fundamental units and of which no material standard need be 
preserved. Thus the unit of area is defined as the area of a 
square of which each side is a unit of length, and there is no 
need of preserving a material standard of the unit of area. The 
unit of velocity is defined as unit distance traveled per second, 

*See Art. 12, on dimensions of derived units. 

|The standard candle is a fundamental unit because it is fixed by an arbitrary- 
preserved standard, and a degree centigrade is a fundamental unit because it is 
preserved in the constant temperature-difference between freezing water and boiling 
water. 



2% ELEMENTS OF MECHANICS. 

and there is no need of preserving a material standard of the unit 
of velocity, indeed, it would be impracticable to preserve a 
velocity. 

Remark I. Quantities such as area, volume, velocity, electric 
current, etc., for which derived units are used, may be called de- 
rived quantities for the reason that they are denned (as quantities) 
in terms of the fundamental quantities, length, mass, and time. 
For example, the density of a body is defined as the ratio of its 
mass to its volume; the velocity of a body is denned as the 
quotient obtained by dividing the distance traveled during an in- 
terval of time, by the interval, etc. 

Remark 2. The choice of fundamental units is a matter 
which is governed solely by practical considerations; in the first 
place the fundamental units must be easily preserved as material 
standards, and in the second place the fundamental quantities 
must be susceptible of very accurate measurement, for the defini- 
tion of a derived unit cannot be realized* with greater accuracy 
than the fundamental quantities can be measured. 

11. The c. g. s. system of units, f Derived units based upon 
the centimeter as the unit length, the gram as the unit mass, and 
the second as the unit time, are in common use. This system of 
derived units is called the c. g. s. (centimeter-gram-second) 
system. 

Thus the square centimeter is the c. g. s. unit of area, the 
cubic centimeter is the c. g. s. unit of volume, one gram per 
cubic centimeter is the c. g. s. unit of density, one centimeter per 
second is the c. g. s. unit of velocity, etc. 

*The definition of a physical quantity is always an actual physical operation. 
Thus the mass of a body is defined by the operation of weighing with a balance; the 
density of a body is defined by the operation involved in the finding of mass and 
volume, for mass and volume must be determined before mass can be divided by 
volume to give density. 

fSo long as the English units of length and mass continue to be used, it will be 
necessary for engineers to use the units of the f . p. s. (foot-pound-second) system to 
some extent, although these systematic f. p. s. units are, many of them, never used 
in commercial work. Thus the f. p. s. unit of force is the poundal, the f. p. s. 
unit of work is the foot-poundal. 



PHYSICAL ARITHMETIC. 2g 

Practical units. In many cases the c. g. s. unit of a quantity 
is either inconveniently small or inconveniently large so that the 
use of the c. g. s. unit would involve the use of very awkward 
numbers. Thus the power required to drive a small ventilating 
fan is 500,000,000 ergs per second, the electrical resistance of 
an ordinary incandescent lamp is 220,000,000,000* c. g. s. 
units of resistance, the capacity of an ordinary Leyden jar is 
0.000,000,000,000,000,005 c. g. s. units of capacity. In such 
cases it is convenient to use a multiple or a sub-multiple of the 
c. g. s. unit as a practical unit. Thus 5 X io 8 ergs per second 
is equal to 50 watts, 22 X io 10 c. g. s. units of resistance is 220 
ohms, 5 X io -18 c. g. s. units of electrostatic capacity is equal to 
0.005 micro-farad. 

Legal units. The system of units now in general use presents several cases in 
which the fundamental measurement of a derived quantity in terms of length, mass, 
and time is extremely laborious and not very accurate at best. Thus the measure- 
ment of electrical resistance in terms of length, mass, and time is very difficult, 
whereas the measurement of electrical resistance in terms of the resistance of a given 
piece of wire is very easy indeed and it may be carried out with great accuracy. In 
every such case the fundamental measurement is carried out once for all with great 
care and the best possible material copy is made of the derived unit and this copy 
is adopted as the standard legal unit. 

12. Dimensions of derived units. The definition of a derived 
unit always implies an equation which involves the derived unit 
together with one or more of the fundamental units of length, 
mass, and time. This equation solved for the derived unit is 
said to express the dimensions of that unit, f Thus the velocity 
of a body is defined as the quotient l/t, where I is the distance 
traveled by the body during the interval of time t, so that the unit 
of velocity is equal to the unit of length divided by the unit of 
time. 

Examples. Let I be the unit of length, m the unit of mass, 
and t the unit of time. Then the unit of area is equal to Z 2 , the 

*In the writing of very large or very small numbers it is always more convenient 
and more intelligible to use a positive or negative power of io as a factor. Thus 
220,000,000,000 is be 
written as 5 X io -18 . 

fAnd also the dimensions of the derived quantity. 



30 ELEMENTS OF MECHANICS. 

unit of volume is equal to Z 3 , the unit of density is equal to m/l 3 
the unit of velocity is equal to l/t, the unit of force is equal to 
ml/t 2 , the unit magnetic pole is equal to V mf\t, etc. 

Naming of derived units. Many derived units have received 
specific names. Such are the dyne, the erg, the ohm, the ampere, 
the volt, etc. Those derived units which have not received spe- 
cific names are specified by writing, or speaking-out, their dimen- 
sions. Thus the unit of area is the square centimeter, the unit 
of density is the gram per cubic centimeter, the unit of velocity is 
the centimeter per second, the unit of momentum is the gram- 
centimeter per second (written gr. cm. /sec). In the case of units 
which have complicated dimensions this method is not con- 
venient in speech. Thus we specify a certain magnetic pole as 
150 gri cm.^/sec. (spoken, 150 c. g. s. units pole). Some units 
have dimensions so simple that to specify them in terms of their 
dimensions is almost meaningless. Thus an angle has zero dimen- 
sions, being a length divided by a length (an arc divided by a 
radius). An angular velocity is an angle divided by time, and 
dimensionally an angular velocity is equal to i/t. In such cases 
it is impossible to name a unit in terms of its dimensions. 

13. Scalar and vector quantities. A scalar quantity is a quan- 
tity which has magnitude only. Thus everyone recognizes at 
once that to specify 10 cubic meters of sand, 25 kilograms of 
sugar, 5 hours of time, is, in each case, to make a complete spec- 
ification. Volume, mass, time, energy, electric charge, etc., are 
scalar quantities. 

A vector quantity is a quantity which has both magnitude and 
direction, and to specify a vector one must give both its magni- 
tude and direction. This necessity of specifying both the magni- 
tude and direction of a vector is especially evident when one is 
concerned with the relationship of two or more vectors. Thus if 
one travels a stretch of 10 kilometers and then a stretch of 5 kilo- 
meters more, he is by no means necessarily 15 kilometers from 
home; his position is, in fact, indeterminate until the direction of 
each stretch is specified. If one man pulls on a car with a force 



PHYSICAL ARITHMETIC. 3 I 

A of 200 units and another pulls with a force B of 100 units, the 
total force acting on the car is by no means necessarily equal to 
300 units. In fact, the total force is unknown both in magnitude 
and direction until the direction as well as the magnitude of 
each force A and B is specified. Length, velocity, accelera- 
tion, momentum, force, magnetic field intensity, etc., are vector 
quantities. 

Representation of a vector by a line. In all discussions of 
physical phenomena which involve the relationships of vectors, it 
is a great help to the understanding to represent the vectors by 
lines. Thus in the discussion of the combined action of several 
forces on a body, it is a great advantage to represent each force 
by a line. To represent a vector by a line, draw the line in the 
direction of the vector (from any convenient point) and make the 
length of the line proportional to the magnitude of the vector. 
Thus if a northward velocity of 600 centimeters per second of a 
moving body is to be represented by a line, draw the line towards 
the north and let each unit length of line represent a chosen 
number of units of velocity. 

When a vector a is represented by a line, the line is parallel to 
a and the value of a is given by the equation 

a = S>1 

in which I is the length of the line and »S is the number of units 
of a represented by each unit length of the line. The quantity 
S is called the scale to which the line represents the vector a. 

14. Addition of vectors. The addition polygon. Many cases 
arise in physics where it is necessary to consider the single 
force which is equivalent to the combined action of several given 
forces; where it is necessary to consider the single actual velocity 
which is equivalent to several given velocities each produced, it 
may be, by a separate cause; where it is necessary to consider the 
single actual intensity of a magnetic field due to the combined 
action of several causes each of which alone would produce a 
magnetic field of given direction and intensity; and so on. The 



32 



ELEMENTS OF MECHANICS. 



single vector is in each case called the vector-sum, or resultant, of the 
several given vectors. Scalar quantities are added by the ordi- 
nary methods of arithmetic, thus 10 pounds of sugar plus 15 
pounds of sugar is 25 pounds of sugar; but the addition of 
several vectors is not an arithmetical operation, it is a geomet- 
rical operation, and it is for this reason that the addition of vectors 
is sometimes called geometric addition. 

The simplest vector quantity is the movement of a body over 
a given distance in a given direction. Such a movement is called 

a displacement of the body. Thus the 
arrow n in Fig. 1 represents a given 
northward displacement and the arrow 
e represents a given eastward dis- 
placement. If a body moves north- 
ward as represented by the arrow n 
and then eastward as represented by 
the arrow e (or if it moves eastward as 
represented by the arrow e and then northward as represented by 
the arrow n) it will reach the point P. Therefore the displace- 
ment OP is spoken of as the vector sum or resultant of the 
two displacements n and e.* 

Addition of two forces. The parallelogram of forces. Let the 
lines a and b, Fig. 2, represent two forces acting upon any body, 




Fig. 1. 




Fig. 2. Fig. 3. 

a boat for example. The vector-sum, or resultant, of the two 
forces a and b is represented by the diagonal r of the parallelo- 
gram of which a and b are the sides. It is evident that the 

*The arrows n and e in Fig. i are drawn at right angles to each other for the sake 
of simplicity. The above statement is true, however, whatever the angle between 
n and e may be. 



PHYSICAL ARITHMETIC. 



33 



geometric relation between a, b and r is completely represented 
by the triangle in Fig. 3, in which the line which represents the 
force b is drawn from the extremity of the line which represents 
the force a. 

Experimental verification of the parallelogram of forces * The 
experimental verification of the parallelogram of forces is usually 
given as a laboratory exercise in elementary physics. Figure 4 




Fig. 4. 



Fig. 5- 



shows an arrangement which is sometimes used for this purpose. 
Two known forces A and B are made to act at the point P, and 
their resultant C is known to be upwards and equal to the 
weight W. 

Addition of any number of forces. The force polygon. Given 
a number of forces a, b, c and d. Draw the line which represents 
the force a from a chosen point 0, Fig. 5, draw the line which 
represents the force b from the extremity of a, draw the line which 
represents c from the extremity of b, and draw the line which 
represents the force d from the extremity of c. Then the line 
from to the extremity of d represents the geometric sum of the 
forces a, b, c and d. 

*It is possible to reduce the addition of two simple vectors like two forces or two 
velocities to the addition of two displacements. The argument involved in this 
reduction constitutes a mathematical proof of the parallelogram of forces. One 
form of this mathematical proof is given in Art. 60. 



34 ELEMENTS OF MECHANICS. 

To prove the above statement concerning the force polygon 
consider a number of forces which are to be added, add forces 
No. i and 2 by using the parallelogram of forces, add No. 3 to the 
resultant of No. 1 and 2 in the same way, add No. 4 to the re- 
sultant of No. 1, 2 and 3 in the same way, and so on, and the 
truth of the above statement will appear at once in the diagram 
so constructed. 

The vector sum of a number of forces is equal to zero if the forces 
are parallel and proportional to the sides of a closed polygon,* the 
directions of the forces being in the directions in which the sides 
of the polygon would be traced in going round the 'polygon. 

A particular case of this general proposition is that the vector 
sum or resultant of three forces is equal to zero if the three forces 
are parallel and proportional to the three sides of a triangle and 
in the directions in which the sides would be passed over in going 
round the triangle. 

Note. What is said above concerning the addition of forces applies to the 
addition of vectors of any kind, velocities, accelerations, magnetic field intensities 
and so on. 

15. Resolution of vectors. Any vector may be replaced by a 
number of vectors of which it is the vector sum, or in other words 
y-axis a given vector can be resolved 

into parts. The simplest case 
^^i is that in which a vector is re- 

^ ^J placed by two vectors; the 

)P°dj/l^v * wo vec t° rs must be parallel 
^■^ and proportional to the re- 
x-axis spective sides of a parallelo- 
gram of which the diagonal 
represents the given vector. 
Flg ' 6 * The following discussion of 
the resolution of forces illustrates this matter. 

Consider force F in Fig. 6. The two forces X and Y would 
have exactly the same effect as the force F and therefore they 

*The sides of a force polygon need not all lie in a plane except, of course, when 
the polygon is a triangle. 



PHYSICAL ARITHMETIC. 



35 



may be thought of in place of the force F. The two forces X 
and Y in Fig. 6 are called the rectangular components of the force 
F because the angle between X and Y is a right angle. Or, in 
other words, if a rectangle be constructed whose diagonal repre- 
sents a given force, then the sides of the rectangle represent what 
are called the rectangular components of that force in the direc- 
tions of those sides. 

Example. Consider the force F, Fig. 7, which is pulling on a 
car, the car being constrained to move along the track. The 
force F is equivalent to the two forces a and /? combined. The 




track 



Fig. 7. 



force ,8 has no effect in moving the car (it may have an indirect 
effect on the motion of the car by pulling the flanges of the wheels 
against the rails and thus increasing the friction) so that the 
force o. is the part of F which is effective in moving the car. 
This force oc is called the resolved part of F, or the component of F 
in the direction of the track. 

16. Scalar and vector products and quotients. — The product, or quotient, of a 
scalar and a vector, a, is another vector parallel to a. 

Examples. — The distance I traveled by a body in time t is equal to the product 
vt, where v is the velocity of the body; and I is parallel to v. 

The force F with which a fluid pushes on an exposed area a is equal to the prod" 
uct pa where p is the hydrostatic pressure of the fluid (a scalar). The vector 
direction of an area is the direction of its normal and this is parallel to F. 

A force F which does an amount of work W in moving a body a distance d (which 
is parallel to F) is equal to W/d. 

The acceleration of a body is equal to &v/At where &v is the increment of velocity 
in the time interval &t; the acceleration is a vector which is parallel to &v. 

Vector products. Case I. Parallel vectors. The product or quotient of 
parallel vectors is a scalar. Thus W= Fd, in which W is the work done by a force 
F acting through a distance d in its direction; p = FJ a, in which p is the pressure 



36 



ELEMENTS OF MECHANICS. 



in a liquid which exerts a force F on an exposed area a; V = la, in which V is the 
volume of a prism of base a and altitude I. 

Case II. Orthogonal vectors. The product, or quotient, of two mutually per- 
pendicular vectors is a third vector at right angles to both factors. Thus a = lb 
and b = a/l, in which a is the area of a rectangle of length I and breadth b; T = Fl, 
in which T is the moment or torque of a force .F, and Z is its arm. The product of 
a vector and a line perpendicular thereto is called the moment of the vector. 

Case III. Oblique vectors. The product of two oblique vectors consists of two 
parts, one of which is a scalar and the other is a vector. Consider two vectors, o. 
and ft, Fig. 8. Resolve ft into two components, ft' and ft" respectively parallel to 
and perpendicular to «. Then 

aft = a (/?'+/?") = ap'+ap» t 




in which aft' is a scalar and a/3" is a vector. The scalar part of a vector product 
is indicated thus, 5 1 • aft (read scalar-alpha-beta) . The vector part of a vector prod- 
uct is indicated thus, V • aft (read vector-alpha-beta). When V • aft = o, « and ft 
are parallel; when 5* aft = o, a and ft are orthogonal. 

Examples of products of vectors. The area of any parallelogram is equal to V • bl 
where b and I are the two sides of the parallelogram. The work done by a force 
is equal to S • Fd where F is the force and d is the displacement of the point of 
application of the force. The volume of any parallelopiped is equal to S • al where 
a is the area of the base and I is the length of the other edge. The torque action of 
any force is equal to V • Fr where F is the force and r is the lever arm. 

17. Constant and variable quantities. The study of those 
physical phenomena which are associated with unvarying con- 
ditions is comparatively simple, whereas the study of those 
phenomena which are associated with rapidly varying conditions 
is generally very complicated. Thus, to design a bridge is to so 
proportion its members that a minimum amount of material may 
be required to build it, and it is a comparatively simple problem 
to design a bridge to carry a steady load because it is easy to 
calculate the stress in each member due to a steady load, but it 
is an extremely complicated problem to design a bridge to carry 



PHYSICAL ARITHMETIC. 37 

a varying load, such as a moving locomotive which comes upon 
the bridge suddenly and moves rapidly from point to point. 

Two kinds of variations are to be distinguished, namely, varia- 
tions in space and variations in time. 

Variations in time. In the study of phenomena which de- 
pend upon conditions which vary in time, that is, upon conditions 
which vary from instant to instant, it is necessary to direct the 
attention to what is taking place at this or that instant; or, in 
other words, to direct the attention to what takes place during 
very short intervals of time; or, borrowing a phrase from the 
photographer, to make snap-shots, as it were, of the varying 
conditions. 

Definition of rate of change.* Principle of continuity. In 
order to establish the rather difficult idea of instantaneous rate 
of change of a varying quantity, it is a great help to make use 
of a simple physical example. Therefore let us consider a pail 
out of which water is flowing through a hole in the bottom. Let 
x be the amount of water in the pail. Evidently x is a changing 
quantity. Let Ax be the amount of water which flows out of 
the pail during a given interval of time At, then the quotient 
Ax/ At is called the average rate of change of x during the given 
interval of time, and, if the interval At is very short, the quotient 
Ax I At approximates to what is called the rate of change of x at 
a given instant, or the instantaneous rate of change of x. 

♦Nearly everyone falls into the idea that such an expression as 10 feet per second 
means 10 feet of actual movement in an actual second of time, but a body moving at 
a velocity of 10 feet per second, might not continue to move for a whole second, or 
its velocity might change before a whole second has elapsed. Thus, a velocity of 10 
feet per second is the same thing as a velocity of 864,000 feet per day, but a body 
need not move steadily for a whole day in order to move at a velocity of 864,000 
feet per day. Neither does a man need to work for a whole month to earn money at 
the rate of 60 dollars per month, nor for a whole day to earn money at the rate of 2 
dollars per day. A falling body has a velocity of 19,130,000 miles per century after 
it has been falling for one second, but to specify its velocity in miles per century does 
not mean that it moves as far as a mile or that it continues to move for a century! 
The units of length and time which appear in the specification of velocity are com- 
pletely swallowed up, as it were, in the idea of velocity, and the same thing is true 
of the specification of any rate. 



38 ELEMENTS OF MECHANICS. 

The amount of water which flows out of the pail during a short 
interval of time is very nearly proportional to the time ; and if a 
shorter and shorter time interval is considered the amount of 
water which flows out of the pail is more and more nearly in 
exact proportion to the time so that the quotient Ax/ At approaches 
a perfectly definite finite value as At and Ax both approach zero. 
If the amount of water in the pail were to change by sudden 
jumps as it were, then the amount of water flowing out of the 
pail in a very short interval would not be even approximately 
proportional to the interval, and the rate of change of x at a 
given instant would be unthinkable; but physical quantities 
which vary in value from instant to instant always vary con- 
tinuously and have, therefore, at each instant a definite rate of 
change ; that is to say, the quotient Ax/ At always approaches a 
definite limiting value as the interval At is made shorter and 
shorter. 

Let it be understood that this paying attention to what takes 
place during very short intervals of time does not refer to obser- 
vation but to thinking, it is a matter of mathematics,* and there- 
fore the following purely mathematical illustration is a legitimate 
example. 

Example. Consider a square of which the sides are growing 
in proportion to elapsed time so that the length of each side of 

*Two distinct methods are involved in the directing of the attention to what 
takes place during infinitesimal time intervals or infinitesimal regions of space. 

(a) The method of differential calculus. A phenomenon may be prescribed as a 
pure assumption and the successive instantaneous aspects derived from this prescrip- 
tion. Thus we may prescribe uniform motion of a particle in a circular path, and 
then proceed to analyze this prescribed motion as exemplified in Art. 38; or we may 
prescribe a uniform twist in a cylindrical metal rod, and then proceed to analyze the 
prescribed distortion (Art. 135). This method is also illustrated by the example given 
in the text, in which the expression for the growing sides of a square is prescribed. 

(6) The method of integral calculus. It frequently happens that we know the 
action which takes place at a given instant or in a small region, and can formulate this 
action without difficulty, and then the problem is to build up an idea of the result of 
this action throughout a finite interval of time, or throughout a finite region of space. 
For example, a falling body gains velocity at a known constant rate at each instant, 
how much velocity does it gain and how far does it travel in a given finite interval 
of time? 



PHYSICAL ARITHMETIC. 39 

the square may be expressed as kt, where k is a constant and t 
is elapsed time reckoned from the instant when one of the sides is 
equal to zero. Then the area of the square is S = k 2 t 2 , and it is 
evident that the area is increasing. Let t -f- At be written for t 
in the expression for S and we have 

5 + AS = k\t + At) 2 = k 2 t 2 + 2k 2 t-At + k 2 (At) 2 

whence, subtracting 5 = k 2 t 2 , member from member, we have 

AS= 2k 2 t-At + k 2 (At) 2 
or 

-- = 2k 2 t + k 2 At 
At 

from which it is evident that AS/ At becomes more and more 
nearly equal to 2k 2 t as At is made smaller and smaller. The 
value of AS/ At for an indefinitely small value of At is usually 
represented by the symbol* dS/dt so that we have 

dS ,2 

It = 2kl 

Propositions concerning rates of change, (a) Consider a quan- 
tity x which changes at a constant rate a, then the total change 
of x during time t is equal to at. For example, a man earns 
money at the rate of 2 dollars per day and in io days he earns 
2 dollars per day multiplied by io days which is equal to 20 
dollars. A falling body gains 32 feet per second of velocity every 
second, that is, at the constant rate of 32 feet per second per 
second, and in 3 seconds it gains 32 feet per second per second 
multiplied by 3 seconds which is equal to 96 feet per second. 

(b) Consider a quantity y which always changes k times as fast 
as another quantity x, then, if the two quantities x and y start 
from zero together, y will be always k times as large as x. That 

*The symbol dS / dt is one single algebraic symbol and it is not to be treated 
otherwise. It stands for the rate of change of S at any given instant and is to be so 
read. 



40 ELEMENTS OF MECHANICS. 

is, if dy/dt = k • dx/dt, then y = kx if y and x start from zero 
together. 

Conversely, if one quantity is always k times as large as an- 
other it must always change k times as fast. 

(c) Consider a quantity 5 which is equal to the sum of a num- 
ber of varying quantities x, y and z, then the rate of change of 5 
is equal to the sum of the rates of change of x and y, and z. 
This may be shown as follows: let Ax, Ay and Az be the in- 
crements of x, y and z during a given interval of time At, then 

the increment of 5 is 

As = Ax -\- Ay -\- Az K . 

whence, dividing both members by At we have 

As Ax Ay Az 
At = ~At + A7 + At 

or, if the interval At is very short, we have 

ds dx dy dz ,., 

dt dt dt dt 

Conversely, if the relation (i) is given, that is, if 5 is a quantity 
whose rate of change is known to be equal to the sum of the 
rates of change of x, and y, and z, then 5 must be equal to 
oo + y + z if s, x, y, and z all start from zero together. 

Variations in space. Imagine a bar of iron one end of which 
is red hot and the other end of which is cold. Evidently the 
temperature of the bar varies from point to point. The pressure 
in a vessel of water increases more and more with the depth, the 
density of the air decreases more and more with increasing alti- 
tude above the sea. 

Such quantities as temperature, pressure, and density which 
refer to the physical conditions at various points in a substance 
are called distributed quantities. The distribution is said to be 
uniform when the quantity has the same value throughout a sub- 
stance, the distribution is said to be non-uniform when the quan- 



PHYSICAL ARITHMETIC. 4 1 

tity varies in value from point to point. Thus the temperature 
of the air in a room is uniform if it is the same throughout, 
whereas the temperature of a bar of iron which is red hot at one 
end and cold at the other end is non-uniform. 

In the study of phenomena dependent upon conditions which 
vary from point to point in space, the attention must be directed to 
what takes place in very small regions, because too much takes 
place in a finite region. Let it be understood, however, that this 
paying attention to what takes place in small regions does not 
refer to observation but to thinking, it is a matter of mathematics, 
and therefore the following illustration is a legitimate example, 
even though the physics may not be entirely clear: 

A rigid wheel rotates at a speed of n revolutions per second. 
Let us consider what is called the kinetic energy of the wheel. 
Now the kinetic energy in ergs of a moving body is equal to }imv 2 , 
where m is the mass of the body in grams and v is its velocity in 
centimeters per second ; but the difficulty here is that the different 
parts of the wheel have different velocities, and if we are to apply 
the fundamental formula for kinetic energy ( = j4mv 2 ) to a rotating 
wheel it is necessary to consider each small portion of the wheel 
by itself. Thus, a small portion of the wheel at a distance r from 
the axis has a velocity which is equal to 2-wnr, and if we repre- 
sent the mass of the small portion by Am, the kinetic energy 
of the portion will be *4 X Am X (2-irnr) 2 , or 2ir 2 n 2 -r 2 Am', so 
that the total kinetic energy of the wheel will be equal to the 
sum of a large number of such terms as this. But, the factor 
27r 2 ft 2 is common to all the terms, and therefore the total kinetic 
energy is equal to 2ir 2 n 2 times the sum of a large number of terms 
like r 2 Am. This sum is called the moment of inertia of the wheel. 

Gradient. Consider an iron bar of which the temperature is 
not uniform. Let AT be the difference of the temperatures at 
two points distant Ax from each other, then the quotient ATJAx 
is called the average temperature grade or gradient along the 
stretch Ax, and if Ax is very small the quotient ATJAx is called 
the actual temperature gradient and it is represented by the 



42 ELEMENTS OF MECHANICS. 

symbol dT/dx. The use of this idea of temperature gradient is 
illustrated in the discussion of the conduction of heat. 

18. Varying vectors.* The foregoing article refers solely to 
varying scalar quantities. The mathematics of varying vectors 
may also be considered in two parts, namely time variation and 
space variation. 

Time variation of velocity, f The velocity of a body is defined 
as the distance traveled in a given time divided by the time. 
When a velocity always takes place in a fixed direction it may be 
thought of as a purely scalar quantity. Thus -a falling body has 
a velocity of 50 feet per second at a given instant and 3 seconds 
later it has a velocity of 146 feet per second, so that the increase 
of velocity in three seconds in 96 feet per second and the rate of in- 
crease is 96 feet per second divided by 3 seconds which is equal to 
32 feet per second per second; but, suppose that the velocity 
of a body at a given instant is 50 units in a specified direction 
and that 3 seconds later it is 146 units in some other specified 
direction, then the change of velocity is by no means equal to 96 
units and the rate of change of the velocity is by no means equal 
to 32 units per second. 

The rate of change of the velocity of a body is called the 
acceleration of the body. Consider any moving body, a ball 
tossed through the air for example, let its velocity, v v at a given 
instant be represented by the line OA, Fig. ga, let its velocity 
v 2 at a later instant be represented by the line OB, and let the 
elapsed time interval be At. Now the velocity which must be 

*This branch of mathematics is very largely ignored in present undergraduate 
courses, and yet no one can have a clear insight into the phenomena of motion with- 
out having an idea of the time variations of velocity, and no one can have a clear 
insight into the phenomena of fluid motion and of electricity and magnetism without 
having some understanding of the space variation of such vectors as fluid velocity, 
magnetic field, and electric field. 

fWhat is here stated concerning the time variation of velocity applies to the time 
variation of any vector whatever. Thus if any varying vector is represented to 
scale by a line drawn from a fixed point, then the velocity of the end of the line 
represents the rate of change of the vector to the same scale that the line itself rep- 
resents the vector. 



PHYSICAL ARITHMETIC. 



43 



added (geometrically) to v l to give v 2 is the velocity Av which is 
represented by the line AB. Therefore the, change of the ve- 
locity of the tossed ball during the interval At is the vertical 




Fig. ga. 



Fig. 9&. 



velocity Av shown in the figure, and the acceleration, a, of the ball 
is equal to Av/At which is of course in the direction of Av. If 
the varying velocity of a tossed ball be represented by a line OP, 
Fig. gb, drawn from a fixed point 0; then, as the velocity changes 
the line OP will change, the point P will move, and the velocity 
of the point P will represent the acceleration of the body to the 
same scale that the line OP represents the velocity of the body. 




Fig. ioa. 



Fig. io&. 



The orbit of a moving body is the path which the body de- 
scribes in its motion. Thus the orbit of a tossed ball is a para- 
bola as shown in Fig. ioa, and the orbit of a ball which is twirled 



44 



ELEMENTS OF MECHANICS. 



on a cord is a circle as shown in Fig. 11a. Let the line OP, 
Fig. 10b or Fig. lib, drawn from a fixed point, be imagined to 
change in such a way as to represent at each instant the velocity 
of the body as it describes its orbit, then the end P of the line 
will describe a curve called the hodo graph of the orbit, and of 
course, the velocity of the point P will represent at each instant 
the acceleration of the moving body. Thus the hodograph of a 




\ 



\ 



\ 



0*r 



■HP 



/ 






hodograph .S 



Fig. 1 1 a. Fig. n&. 

tossed ball is a vertical straight line as shown in Fig. 10b, this is 
evident when we consider that a tossed ball has a constant ac- 
celeration vertically downwards, so that the point P, Fig. 10b, 
must move at a constant velocity vertically downwards. The 
hodograph of a ball twirled on a string is a circle as shown in 
Fig. 1 1 b, this is evident when we consider that the magnitude of 
the velocity of the body in Fig. i \a is constant, so that the length 
of OP, Fig. lib, which represents the velocity of the body, must 
also be constant. Furthermore, the velocity of the ball is always 
at right angles to the cord in Fig. 11a, and therefore the line 
OP, Fig. lib, is always at right angles to the cord, so that the 
generating point P of the hodograph makes the same number of 
revolutions per second as the twirled ball. It is important to 
note also that the velocity of the point P in Fig. lib, is always 
parallel to the cord in Fig. 11a, that is, the acceleration of the 
ball in Fig. 11a is at each instant in the direction of the cord. 



PHYSICAL ARITHMETIC. 



45 



Space variation of vectors. The simplest idea connected with the space variation 
of a vector is the idea of the stream line, if it may be permitted to use the terminology 
of fluid motion to designate a general idea. A stream line in a moving fluid is a line 
drawn through the fluid so as to be at each point parallel to the direction in which 
the fluid is moving at that point. Thus if a pail of water be rotated about a vertical 
axis, the stream lines are a system of concentric circles as shown in Fig. 12a, and 
Fig. 12b represents the approximate trend of the stream lines in the case of a jet of 
water issuing from a tank. Where the stream lines come close together the velocity 
of the water is great and where they are far apart the velocity is small. 




The potential of a distributed vector. — In some simple cases of fluid motion it is 
geometrically possible to look upon the stream lines as the lines of slope of an 
imagined hill, the steepness of which represents the velocity of the fluid at each 
point both in magnitude and in direction. The height of this imagined hill at a 
point is called the potential of the fluid velocity at that point. The idea of potential 
is especially useful in the study of electricity and magnetism.* 

Problems. 

7. A table top is 10 feet long and 50 inches wide. Find its 
area in inch-feet, and explain the result. Ans. 500 inch-feet. 
One inch-foot is the area of a rectangle one foot long and one 
inch wide. 

8. A body has a mass of 60 pounds and a volume of 2 gallons. 
Find its density without reducing data in any way. Ans. 30 
pounds per gallon. 

9. A water storage basin has an area of 2,000 acres, find the 

*A very simple discussion of the theory of scalars and vectors in space, including 
the theory of potential, is given in Chapter VI of Electric Waves, by W. S. Franklin; 
The Macmillan Co., New York, 1909. 



46 ELEMENTS OF MECHANICS. 

volume of water in acre-feet required to fill the basin to a depth 
of 1 6 feet. Explain the acre-foot as a unit of volume and find 
the number of gallons in one acre-foot. Ans. 32,000 acre-feet. 
The acre-foot is the volume of water required to fill a one-acre 
pond one foot deep. It is equal to 325,800 gallons. 

10. A man travels at a velocity of 6 feet per second; how far 
does he travel in two hours? Find the result without reducing 
the data in any way. Explain the foot-hour per second as a unit 
of length. Ans. Distance traveled is 12 hour-feet-per-second ; 
one hour-foot-per-second is the distance passed over in one hour 
by a man walking at a speed of one foot per second. 

11. A man starts from a given point and walks three miles due 
north, then two miles northeast, then 2 miles south, and then 
one mile east. Show by means of a vector diagram how far, 
and in what direction he is from his starting point. Ans. Draw 
an actual diagram to scale showing the distances traveled. 

12. A stream flows due south at a velocity of two miles per 
hour. A man rows a boat in an eastward direction at a velocity 
of four miles per hour. What is the actual velocity of the boat 
and in what direction is it moving? Ans. 4.472 miles per hour; 
63 ° 26' east of south. 

Note. The actual velocity of the boat is the vector sum of the velocity of the 
boat with reference to the water (an eastward velocity of 4 miles per hour) and 
the velocity of the stream (a southward velocity of 2 miles per hour). 

13. A stream flows due south at a velocity of two miles per 
hour. A man, who can row a boat at a velocity of four miles per 
hour, wishes to reach the opposite bank at a point due east of his 
starting point. In what direction must he row? Ans. 30 up 
stream. 

Note. The actual velocity of the boat is, in this case, to be an eastward velocity, 
and it is equal to the vector sum of the velocity of the boat with reference to the 
water and the velocity of the water. 

From a point O draw a line OA representing the southward velocity of the 
stream. From the point O draw a line OB in a due eastward direction. With 
the point A as a center describe a circle of which the radius represents a velocity 
of 4 miles per hour, and let P be the point where this circle cuts the line OB. Then 
the line AP represents the required direction in which the boat must be rowed. 



PHYSICAL ARITHMETIC. 



47 



14. An anemometer on board ship indicates a wind velocity of 
28 miles per hour apparently from the northeast. The ship, how- 
ever, is moving due north at a velocity of 15 miles per hour. 
What is the actual direction and velocity of the wind? Ans. 
1 3 37' south of west; velocity of wind 20.37 miles per hour. 

Note. The apparent velocity of the wind to a person on shipboard is the vector 
sum of the two parts, namely, (a) the actual velocity of the wind and (&) what may 
be called the boat wind, that is to say, a wind which is due to the velocity of the 
boat alone. This boat wind is equal and opposite to the velocity of the boat. In 
this particular problem the apparent wind of 28 miles per hour from the northeast 
is the vector sum of the actual wind and the boat wind of 1 5 miles per hour towards 
the south. Therefore draw a line OA representing the apparent velocity of the 
wind of 28 miles per hour towards the southwest. Draw another line OB repres- 
senting the boat wind of 15 miles per hour towards the south. Then the line B A 
represents the direction and velocity of the actual wind. 

15. A gun which produces a projectile-velocity of 200 feet per 
second is mounted aboard a car with its barrel G at right angles 
to the direction of motion of the car, as shown in Fig. 1 $p. The 




y. y id y 



50 feet pei? second 
Fig. i$p. 

car is traveling 50 feet per second. The sights are to be arranged 
at an angle 6 to the gun barrel as shown. Find the value of 
so that the ball may hit any object which, at the instant of firing, 
is in the line S of the sights. Ans. = 14 2' 10"; actual 
velocity of projectile is 206 feet per second. 

Note. The actual velocity of the bullet in this problem is the vector sum of the 
velocity of the car and 200 feet per second at right angles to the motion of the car, 
and the sight line must, of course, be in the actual direction in which the bullet is 
moving when it leaves the gun. 

16. A body moves at a velocity of 20 miles per hour in a direc- 
tion 20 north of east; find the northward and eastward com- 



4^ ELEMENTS OF MECHANICS. 

ponents of its velocity. Ans. Northward component 6.84 miles 
per hour; eastward component 18.8 miles per hour. 

17. Find the magnitude and direction of the single force which 
is equivalent to the combined action of three forces A, B and C; 
force A being northward and equal to 200 pounds, force B being 
towards the north-east and equal to 150 pounds, and force C 
being eastwards and equal to 100 pounds. Ans. 370 pounds; 
34 31' east of north. 

18. A horse pulls on a canal boat with a force of 600 pounds- 
weight and the rope makes an angle of 25 ° with the line of the 
boat's keel. Find the component of the force parallel to the keel. 
Ans. 543.8 pounds. 

19. If 500 grams of water leak out of a pail in 26 seconds, 
what is the average rate of leak? Ans. 19.22 grams per second. 

20. A man earns $27.50 in 8*4 days. What is the average 
rate at which he earns money? Ans. $3,235 per day. 

21. A man's wages increase from $50 per month to $150 per 
month in the course of 3 years. What is the average rate of 
increase of wages? Ans. $33/^ per month per year. 

22. During 28 seconds the velocity of a train increases from 
zero to 12 feet per second. What is the average rate of increase 
of velocity? Ans. 0.429 feet per second per second. 

23. A train gains a speed of 32 miles per hour in 80 seconds. 
Find its average acceleration in miles per hour per second. Ans. 
0.4 mile per hour per second. 

24. A pole 22 feet long is dragged sidewise over a field at a 
velocity of 8 feet per second. At what rate does the pole sweep 
over area? Ans. 176 square feet per second. 

25. A prism has a base of 25 square centimeters, and its height 
is increasing at the rate of 5 centimeters per second. How fast is 
its volume increasing? Ans. 125 cubic centimeters per second. 

26. The slope of a hill falls 60 feet in a horizontal distance of 
270 feet. What is the grade? Ans. 0.222 of a foot drop for each 
horizontal foot of distance, or 22.2 per hundred, or 22.2 per cent. 

27. One side of a brick wall is at a temperature of o° C. and 



PHYSICAL ARITHMETIC. 49 

the other side is at a temperature of 23 ° C. The wall is 30.5 
centimeters thick. What is the average temperature gradient 
through the wall? Ans. 0.754 centigrade degrees per centimeter. 
28. At a given point in a water pipe the water pressure is no 
pounds per square inch. Twenty-two feet from this point the 
pressure is 75 pounds per square inch. What is the average pres- 
sure gradient along the pipe? Ans. 1.59 pounds per square inch 
per foot. 



CHAPTER III. 

SIMPLE STATICS * 

19. Balanced force actions. When a body remains at rest, 
or when a body continues to move with uniform velocity along 
a straight line, or when a symmetrical body like a wheel con- 
tinues to rotate at uniform speed about a fixed axis, the forces 
which act on the body are balanced. A very brief discussion of 
this matter is given on pages 5 to 7. The study of balanced 
forces or, as it is sometimes expressed, the study of forces in 
equilibrium, constitutes the science of statics. The science of 
statics is indeed the study of equilibrium in its widest sense, 
including the equilibrium of the forces which act on the parts of 
a distorted body (the statics of elasticity), and the equilibrium 
of the forces which act upon the parts of a fluid (hydrostatics) ; 
but these branches of statics are treated in subsequent chapters. 
The present chapter deals only with sets of forces which do not 
produce translatory motion or rotatory motion, f 

Every one knows that a single force acting on a body may 
cause both translation and rotation. Thus a boat, which is 
pushed away from a landing, generally turns more or less as it 
moves away. In this case, however, it may be the force action 
of the water on the boat that causes the turning; but a sidewise 
push on the bow of the boat certainly produces both translation 
and rotation independently of the force action of the water. 

From the fact that a single force can produce both translation 
and rotation, it may seem as though it would be impossible to 
consider separately the two effects of a force, namely (a) the 
tendency to produce translatory motion, and (b) the tendency to 

*One of the best treatises on statics is Part II of Alexander Ziwet's Theoretical 
Mechanics (The Macmillan Co.). A very complete treatment is G. M. Minchin's 
Treatise on Statics in two volumes. (Clarendon Press.) 

fSee Art. 29 for definitions of translatory motion and rotatory motion. 

5° 



SIMPLE STATICS. 



51 



produce rotatory motion; but every one knows that forces can 
produce translation without producing rotation, and that forces 
can produce rotation without producing translation. Thus, a 
table can be moved to one side without turning, or a table can 
be turned without being moved to one side. The fact is that 
the two tendencies a and b must be considered separately. The 
tendency of a force to produce translatory motion and the ten- 
dency of a force to produce rotatory motion constitute two dis- 
tinct types of force action as explained in Art. 29. 

The tendency of a force to produce translatory motion depends 
only upon the direction and magnitude of the force.* 

The tendency of a force to produce rotatory motion about a 
given axis is called the torque action of the force about that axis 
or the moment of the force about that axis, and it is equal f to the 
product of the force and the perpendicular distance from the 




Fig. 13a. 



Fig. 136. 



axis to the line of action of the force. Thus the torque action 
of the force F about the axis in Fig. 13a is equal to the product 
FV '. The torque action of the force F about the point in Fig. 
13 a. is also equal to the product F'l as shown in Fig. 13&. 



*This statement appears on its face to be contrary to experience as follows: 
Every one knows that a force applied at the middle of a bureau drawer may pull 
the drawer out, whereas the same force applied at one end will not pull the drawer 
out. When the outward pull is applied at the end of the drawer, however, it causes 
the drawer to bind against the guides thus bringing into action a new set of 
forces which do not exist when the outward force is applied at the middle of the 
drawer. 

fThe simplest argument which leads to an idea of the value of a torque is the 
argument which is given in Art. 60. 



52 ELEMENTS OF MECHANICS. 

20. First condition of equilibrium. The condition that must 
be satisfied in order that a number of forces may have no ten- 
dency to produce translatory motion is called the first condition 
of equilibrium and it may be stated in two ways as follows: 

(a) In order that a number of forces may have no tendency to 
produce translatory motion, it is necessary that what is called 
the vector sum* of the forces be equal to zero, that is, the respec- 
tive forces must be parallel and proportional to the sides of a 
closed polygon and in the directions in which the sides of the 
polygon would be passed over in going round the polygon. This 
statement of the first condition of equilibrium leads directly to 
the graphical method of solving problems in statics. 

(b) Imagine each of the forces acting upon a body to be re- 
solved into rectangular components (x, y and s-components) . 
Then the sum of all the x-components must be equal to zero, 
the sum of all the ^-components must be equal to zero, and the 
sum of all the s-components must be equal to zero. In applying 
this condition of equilibrium it is frequently convenient to pick 
out all of the positive x-components (to the right) and all the 
negative ^-components (to the left) and place the sum of the 
forces to the right equal to the sum of the forces to the left; 
and to proceed in a similar way with respect to the y and z- 
components. 

Examples of the application of the first condition of equilibrium. 

Problems in statics which involve no question of rotation are 
solved by applying the first condition of equilibrium. There 
are four cases as follows: (a) When everything is given but the 
magnitude and direction of one force, (b) when everything is 
given but the magnitude of one force and the direction of another, 
(c) when everything is given but the magnitudes of two forces, 
and (d) when everything is given but the directions of two forces. 
Solution of case (a). Starting from a point in Fig. 14a, 
draw a line representing the known direction and magnitude of 
force No. 1. From the end of this line draw a line representing 

*See Art. 14. 



SIMPLE STATICS. 



53 



the known direction and magnitude of force No. 2. From the 
end of this line draw a line representing the known direction 
and magnitude of force No. 3, and so on. Let P be the point 





Fig. 14a. 



Fig. 14&. 



ultimately so reached. Then the line PO represents the un- 
known force in magnitude and direction. Figure 14& shows the 
forces as they appear when acting on the body. 

Solution of case (b). Starting from a point 0, Fig. 15a, draw a 
line representing the known direction and magnitude of force 
No. 1. From the end of this line draw a line representing the 
known direction and magnitude of force No. 2, and so on. Let 

\ 

\ 

\ 





Fig. 15a. 



Fig. 15&. 



P be the point ultimately so reached. From P as a center de- 
scribe a circle of which the radius represents the known value of 
force G, and from draw a line in the known direction of force H. 



54 



ELEMENTS OF MECHANICS. 



The problem has two solutions; the forces corresponding to one 
solution are I, 2, G' and H' '; and the forces corresponding to 
the other solution are 1, 2, G" and H" . Figure 156 shows the 
forces 1, 2, G' and H' as they appear when acting on the body. 





Fig. 16a. 



Fig. 16b. 



Solution of case (c). Starting from a point 0, Fig. 16a, draw 
a line representing the known direction and magnitude of force 
No. 1. From the end of this line draw a line representing the 
known direction and magnitude of force No. 2, and so on. Let 





Fig. 17a. 



Fig. 17&. 



P be the point ultimately so reached. Through the point P 
draw a line in the known direction of force G and through the 
point draw a line in the known direction of force H. The 
lines G and H in Fig. 16a represent the required magnitudes of 



SIMPLE STATICS. 



55 



the two forces. Figure 16b shows the forces i, 2, G and H as 
they appear when acting on the body. 

Solution of case (d). Starting from a point 0, Fig. 17a, draw 
a line representing the known direction and magnitude of force 
No. 1. From the end of this line draw a line representing the 
known direction and magnitude of force No. 2, and so on. Let 
P be the point ultimately so reached. With P as a center de- 
scribe a circle of which the radius represents the known magnitude 
of force G, and with as a center describe a circle of which the 
radius represents the known magnitude of force H. This prob- 
lem has two solutions; the forces corresponding to one solution 
are represented by the lines 1, 2, G' and H' ; and the forces 
corresponding to the other solution are represented by the lines 
1, 2, G" and H". Figure ijb shows the forces 1, 2, G' and H f 
as they appear when acting on the body. 

Remark. The above discussion of the four cases, (a) (b) (c) 
and (J), is based upon the first form of statement of the first 
condition of equilibrium. To show that the second form of state- 
ment leads to the same result let us 
consider case (a) in which the forces 
1, 2, 3 are given and the force G is 
to be found. Figure 18 shows the 
^-components of the forces 1, 2, 3 
and G, and it is evident from the 
figure that X 1 + X 2 = X s + X G 
from which X G may be calculated. 
In the same way the ^-component 
of the unknown force G may be 
calculated. 

The identity of the first and second statements of the first 
condition of equilibrium may be appreciated if one considers that 
the sum of the x-components of the sides of a closed polygon is 
equal to zero, ^-components to the right being considered 
as positive and x-components to the left being considered 
as negative. 




Fig. 18. 



56 



ELEMENTS OF MECHANICS. 



21. Second condition of equilibrium. In order that a number 
of forces may have no tendency to turn a body about a chosen 
axis it is necessary that the sum of the torque actions of the 
forces about that axis be equal to zero; torques tending to turn 
the body in one direction being considered as positive and torques 
tending to turn the body in the opposite direction being con- 
sidered as negative. The chosen axis (or point) is called the 
origin of moments. In applying the second condition of equilib- 
rium it is usually convenient to pick out all of the torques which 
tend to turn the body in a clock- wise direction, and all of the 
torques which tend to turn the body in a counter-clock- wise 
direction, and to place the sum of the clock-wise torques equal 
to the sum of the counter-clock-wise torques. 

Examples of the application of the second condition of equilib- 
rium. Problems in statics which refer to a body mounted upon 
a fixed axis involve no question of translation, and they are 
solved by using the second condition of equilibrium. Thus one 
point of a lever, the fulcrum, is fixed in position, and the equilib- 
rium of the lever depends upon balanced torque actions about 
the fulcrum. 

First example. Consider the two forces A and B which act 
upon a lever as shown in Fig. 19a. The torque action of the force 
A about the fulcrum is equal to Aa, and the torque action of 
the force B about the fulcrum is equal to Bb. These two 




Fig. 19a. 



Fig. 19&. 



SIMPLE STATICS. 



57 



torques are opposite in direction and therefore to balance each 
other they must be equal in value; or, in other words, we must 

have 

Aa = Bb 

Second example. Figure 196 represents a wheel and axle, the 
radius of the wheel being a and the radius of the axle being b. 
The torque action of the force A about the axis is equal to A a, 
and the torque action of the force B about the axis is equal to 
Bb. These two torques are opposite in direction and therefore 
to balance each other they must be equal in value ; or, in other 

words, we must have 

Aa = Bb 

22. Examples of the application of first and second conditions 
of equilibrium. First example. Figure 20a represents a bell- 



2oo 




Fig. 20a. 



crank lever of which the fulcrum is at the point 0. It is required 
to find the value of the force F, and to find the value and direction 
of the force which acts upon lever at 0. Applying the second 
condition of equilibrium we have 

F X 18 inches = 200 pounds X 24 inches X cos 30 

The first member of this equation is the torque action of the force 
F about 0, and the second member is the torque action of the 



58 



ELEMENTS OF MECHANICS. 



200-pound weight about 0. This equation determines the value 
of the force F. 

To determine the force which acts upon the lever at 0, draw a 
line PQ in Fig. 20b representing the known value and direction 
of the force F, and from the end of this line draw the line QR 
representing the known force W in Fig. 20a. The line RP then 
represents in magnitude and direction the force which acts upon 
the lever at 0. 

Second example. A table drawer 36 inches in breadth and 18 
inches in depth (front to back) is pulled by a force F applied at 
a distance x from one corner as shown in Fig. 21. The drawer 
binds at the two corners p and q, and it is required to find the 




Fig. 21. 

smallest value of x for which the drawer can be pulled out by 
the force F, the coefficient of friction* between drawer and guides 
being 0.7. 

The guides exert forces upon the drawer at the two corners p 
and q, and when the drawer is on the point of sliding the force 
exerted on the drawer at p has the two components H and pH 
as shown, and the force exerted on the corner at q has the two 
components G and f*G as shown, /* being the coefficient of friction 
between drawer and guides. 

From the first condition of equilibrium the sum of all the 

*See Arts. 51 and 52 for a discussion of friction. 



SIMPLE STATICS. 



59 



forces to the right must be equal to the sum of all the forces 

to the left, that is 

G = H (i) 

and also the sum of all the downward forces must be equal to 
the sum of all the upward forces in the figure ; that is 

F = pG + pH (ii) 

Choosing the origin of moments at the point q, we have 



Right-handed torques. Left-handed torques. 

F (b - x) = bfiH + aH 



(iii) 



in which a is the depth of the drawer front to back, and b is its 
width. 

All three unknown forces F, G and H may be eliminated from 
these equations and the value of x determined in terms of b, a 
and fJL. 

23. Pure torque. A number of forces which act on a body may 
have no tendency to produce translatory motion and still have a 
tendency to produce rotation, or, in other words, a number of 




Fig. 22. 

forces may satisfy the first condition of equilibrium and not 
satisfy the second condition. Such a combination of forces con- 
stitutes a pure torque and the total torque action is the same 
about any point whatever. For example, the two equal and 
opposite forces which are exerted on the handle of an auger, as 
shown in Fig. 22, constitute a pure torque. Such a pair of forces 
is sometimes called a couple. 



6o ELEMENTS OF MECHANICS. 

24. Three forces in a plane must intersect at a point if they 
are in equilibrium, except when the forces are parallel. Choose 
the origin of moments at the intersection of the lines of action of 
two of the forces. The torque actions of these two forces about 
the origin is then equal to zero, and if the third force does not 
pass through the origin it will have an unbalanced torque about 
the origin and the forces will not be in equilibrium. 

When a problem in statics refers to the equilibrium of three 
forces the second condition of equilibrium is completely ac- 
counted for, insofar as it has a bearing on the problem, by con- 
sidering that the lines of action of the three forces must intersect 
at a point. 

Example. Extend the lines of action of the forces W and F 
backwards in Fig. 20a until they intersect at a point T (point T 
not shown in the figure) . The line of action of the force which is 
exerted on the lever at is the line OT. Having thus deter- 
mined the direction of the force at 0, and knowing the direction 
of the force at F in Fig. 20a, the problem of finding the value of 
the force at is reduced to case (c) in Art. 20. 

25. Resultant of a number of forces. Any number of forces 
(not in equilibrium) which act on a body are together equivalent 
to a single force which is called their resultant; except when the 
forces constitute a pure torque. 

Proof. Given a number of forces in equilibrium. If one of 
these forces (F) is omitted, the combined action of the others 
must be a force equal and opposite to F and having the same line 
of action as F. The exception is also evident since by omitting 
one of a set of forces in equilibrium, the others cannot con- 
stitute a pure torque. 

The magnitude and direction of the resultant of a number of 
given forces is found by taking the vector sum of the forces. 
The point of application of this resultant is a point about which 
the given forces have no torque action. 

First example. Three well matched horses are hitched to an 
evener EE' as shown in Fig. 23a. Horse 1 exerts a certain force 



SIMPLE STATICS. 



61 



A at the point E, and horses 2 and 3 exert at the point E r a 
force B which is equal to 2 A. The forces A and B are parallel, 
and the value of their resultant R is therefore equal to their 
numerical sum. The point of application of the resultant R is 
the point about which the torque actions of A and B balance 
each other. Therefore the forces exerted by the three horses 
are together equivalent to a single force R acting at the point 0. 
Second example. Figure 236 shows two given forces A and B 
acting at the ends of a bar. The resultant of the two forces is 



A E 




horse 1 













\b 






horse 2 


B w b 






7 \ 
E 




► — 


horse 3 



Fig. 23a. 

represented in direction and magnitude by the diagonal R r , and 
the point of application of the resultant may be at any point 
on the line PR'R. The construction of the figure is sufficiently 
evident without further explanation. 

26. Center of gravity.* The center of gravity of a body is the 
point of application of the total force with which the earth pulls 
the body. Thus the total force with which the earth pulls a 
uniform bar may be thought of as applied at the center of the bar. 

♦Considering the forces of gravity on the various parts of a body to be accurately 
parallel, the center of gravity of a body is the same thing as the center of mass of 
the body. 



62 



ELEMENTS OF MECHANICS. 



p 






A\ 






y \ v 










. N \ / N 






' N> \ / 


\ 




\ \ / 


\ 




\ \ / 


\ 




N <v 


\ 


\ 




Fig. 23&. 

Example. A uniform board 16 feet long and weighing 25 
pounds is held in a horizontal position with one end resting on a 
table, and a point 6 feet from the other end resting on the hand. 
It is required to find the forces T and H with which the table 
and hand respectively push upwards on the board. From the 
first condition of equilibrium we have 

T + H = 25 pounds (i) 

Choosing the origin of moments at the center of the board, the 
second condition of equilibrium gives 

T X 8 feet = H X 2 feet (ii) 

The first member of this equation is the torque action of the 
upward push of the table about the origin of moments (center of 
the board), and the second member is the opposite torque action 
of the upward push of the hand about the center of the board. 
These two equations enable the calculation of T and H. 

27. Action and reaction.* A rope pulls on a canal boat and 
the canal boat pulls backwards on the rope with an equal force. 

*This matter is discussed very briefly on page 5. 



SIMPLE STATICS. 63 

A body resting on a table pushes downwards on the table and 
the table pushes upwards on the body with an equal force. The 
mutual force action between the two bodies A and B always consists 
of two equal and opposite forces one of which acts on A and the 
other on B. It is of the utmost importance in the consideration 
of any problem in statics to think of the forces which act on a 
certain body, because a system of forces must act upon one body 
to constitute a system of forces in equilibrium; to think of the 
forces with which the body under consideration acts on other 
bodies leads to confusion. A body rests upon a table; 
the earth pulls downward on the body and of course 
the body pushes downward on the table with an 
equal force. These two forces do not, however, 
constitute a system in equilibrium, because one of 
them acts on the body and the other acts on the 
table; the pull of the earth on the body is balanced 
by the upward push of the table on the body. 

First example. A man weighing 180 pounds is 
seated in a stirrup which is suspended by the rope A Fi 
as shown in Fig. 24a. It is required to find the force 
with which the man must pull downwards on rope B to lift 
himself. Let F be the downward pull of the man on rope B. 
The reaction is an upward pull of rope B on the man equal to 
F. At the same time rope A pulls upwards on the man with a 
force F, so that the total upward pull on the man is 2F; there- 
fore F is equal to 90 pounds. 

Second example. When a problem in statics refers to a mechan- 
ism consisting of several pieces, like a train of gears, the principle 
of equality of action and reaction must be used in discussing the 
equilibrium of the successive parts of the mechanism. Thus 
Fig. 246 shows two levers LL and V V ' . A force F is exerted 
upon the lever L, the other end of this lever exerts a force H 
upon the lever V ', and the other end of the lever L' supports a 
weight W. It is required to find the weight W, having given 
the value of the force F and the two ratios A /a and B/b. The 



64 ELEMENTS OF MECHANICS. 

torque action of the force F about the fulcrum is balanced by 
the torque action of the force G with which the lever L' acts upon 
the lever L. Therefore G = F X A/a. Having thus calculated 
the force G, the force H is known (equal and opposite to G), 
and the value of W is given by the equation W = H X B/b. 

L r onL 

A 

I 

i 

I 






i 

I 

! B 

i 

y 

L anil 

Fig. 246. 

The principle of virtual work. If the forces which act on a body are in equi- 
librium, the total work done by the forces during any small movement of the body 
would be equal to zero. This is called the principle of virtual work. It is discussed 
and illustrated in Chapter V. 

D'Alembert's principle. The ease with which the relationship between a number 
of forces in equilibrium can be shown, especial^ by the use of graphical methods, 
makes it desirable to extend the idea of balanced forces to the subject of dynamics. 
The principle on which this can be done was first enunciated by D'Alembert. The 
following is a statement of D'Alembert's principle as applied to a particular case: 
A part of a mechanism moves in a prescribed manner under the combined action 
of a number of forces; if fictitious forces which are equal and opposite to the forces 
required to produce the known accelerations be introduced into the system, then 
the system of forces (including the fictitious forces) will be in equilibrium. D'Alem- 
bert's principle is exemplified by problems 54 and 56 on pages 112 and 113. 

Problems.* 
29. The sail SS, Fig. 2gp of a sailboat is assumed to be a smooth 
plane, and the force exerted on the sail by the wind, neglecting 

*See Problems in Statics, by Franklin and MacNutt; a collection of problems 
in statics with very complete notes as to methods of solution. Published in pam- 
phlet form by the Macmillan Company. 




SIMPLE STATICS. 6$ 

the slight tangential force (parallel to plane of sail) exerted by the 
wind as it slides or glances along the sail, is at right angles to the 
plane of the sail SS. This force is equal 
to 200 pounds. Find its components par- 
allel to and at right angles to the cen- 
ter-board BB. Ans. 100 pounds parallel 
to center-board; 173.2 pounds at right ">iW ^ 
angles to center-board. 

Note. The component of the force at right angles 
to the center-board BB is counteracted by the pres- 
sure of the water against the center-board, and this 
component produces no perceptible motion of the boat 
because of the large area of the center-board. The Fig- 2gp. 

co mponent of the force which is parallel to the cen- 
ter-board BB is the force which propels the boat forwards. 

30. A ball weighing 100 pounds is suspended by a rope A . The 
ball is pulled to one side by a second rope B, the direction of rope 
B is 30 below the horizontal and the tension of rope B is 60 
pounds. Find the direction and tension of rope A. In answer 
to this problem make a diagram in which three lines, drawn 
outwards from a point, represent the magnitudes and directions 
of the three forces acting on the ball. Ans. 140 pounds; 68° 13' 
above the horizontal. 

31. A ball weighing 100 pounds is suspended by a rope A. 
The ball is pulled to one side by a second rope B which is tied to 
it. Find the two possible directions of rope B for which its 
tension is 60 pounds, and find the tension of rope A, the angle 
between rope A and the vertical being 30 . This problem has 
two answers. For each answer make a diagram in which three 
lines, drawn outwards from a point, represent the magnitudes and 
directions of the three forces acting on the ball. Ans. 63 31' 
above the horizontal; 53.45 pounds. 3 45' above the horizontal; 
119.75 pounds. 

32. A ball weighing 2 pounds hangs by a string and is pushed 
to one side by a strong wind (horizontal) so as to give an angle 
of 30 between the string and the vertical line. Find the tension 
6 



66 



ELEMENTS OF MECHANICS. 



of the string and the force exerted on the ball by the wind. In 
answer to this problem make a diagram in which three lines, 
drawn outwards from a point, represent the magnitudes and 
directions of the three forces acting on the ball. Ans. Tension in 
string 2.31 pounds; force of wind 1.15 pounds. 

33. A ball weighing 100 pounds is suspended by a rope A and 
the ball is pulled to one side by a second rope B which is tied to 
it. Find the directions of ropes A and B in order that the tension 
of A may be 80 pounds and the tension of B 60 pounds. Make 
a diagram in which three lines, drawn outwards from a point, 
represent the three forces acting on the ball.' Ans. 53 8' above 
the horizontal. 

34. A rope 15 feet long stretched between two supports SS 
carries a 1000-pound weight as shown in Fig. 34^. Find the 
tension in each part of the rope, and find the vertical force on each 



14 feet 




Xpoo pounds 

Fig. 34P- 

support, neglecting weight of rope. Ans. 1372 pounds in 5-foot 
section; 1230 pounds in 10-foot section; 691 pounds at right-hand 
support ; 309 pounds at other support. 

35. A simple bridge truss consists of two struts and a tie rod, 
as shown in Fig. 35^. A weight W of 2000 pounds hangs from 
the point P. Find the compression in each strut, the vertical 
pressure on each abutment, and the tension in the tie rod, neglect- 
ing weights of the parts of the truss. Ans. Compression in 10- 
foot strut 1720 pounds. Compression in 13 -foot strut 1450 
pounds. Tension in tie rod 1220 pounds. Vertical force at 
R, 1 21 3 pounds. Vertical force at Q, 787 pounds. 



SIMPLE STATICS. 



6/ 



36. A forty-foot beam, arranged as shown in Fig. ^6p, supports 
a weight of 2000 pounds. Find the pull of the rope and the 





Fig. 35P- 



Fig. 36^. 



thrust of the beam. Ans. Tension in rope 1645 pounds. Com- 
pression in beam 3420 pounds. 

37. The two links of a toggle joint are each 10 inches long be- 
tween centers of end pins, and the center of the middle pin P 




in Fig. up is 0.5 inch from the center line of the end pins (the 

dotted line in Fig. 37^). Find the horizontal components A A of 

the forces exerted on the end pins when the force F is 50 pounds, 

ignoring friction. Ans. 499.4 pounds. 

Note. To neglect the friction is to consider the forces exerted at P to be 
parallel to the respective links of the toggle. 




Fig. 38^. 



68 



ELEMENTS OF MECHANICS. 



38. The span of wire on a telegraph line is 200 feet long, and 
the sag of the line is such that the two lines TT, Fig. 3 Sp, intersect 
at a point three feet below the level of the insulators on the 
poles, the two lines TT being tangent to the wire at the insula- 
tors. The total weight of the wire in the span is 15 pounds. 
Find the tension of the wire at the insulators, find the tension of 
the wire at the center of the span, and find the total downward 
force upon an insulator. Ans. Tension of wire at insulators 
250.11 pounds; tension at center 250 pounds. Downward force 
at insulator 7.5 pounds. 

Note. To solve this problem consider one-half of the span of wire A as a rigid 
body in equilibrium. The force with which the insulator pulls on this body is equal 
and opposite to the tension of the wire at the insulator. To apply the second form 
(b) of the first condition of equilibrium, consider that the force pulling to the left 
on the half-span A is the tension of the wire at the center of the span, and that the 
force pulling to the right on the half-span of wire is the horizontal component of 
the tension of the wire at the insulator; and consider that the force pulling the 
half-span of wire upwards is the vertical component of the tension of the wire at 
the insulator and that the force pulling the half-span of wire downwards is the 
weight of the wire in the half-span. 

39. The cable of a suspension bridge is anchored to a block of 
masonry as shown in Fig. 39^. The block is in the form of a 

cube the length of edge of which 
is 40 feet, the cable being assumed 
to pass over the sharp edge of the 
cube at C. The weight of the ma- 
sonry is 160 pounds per cubic foot. 
Required the tension in the cable 
which will just suffice to turn the 
block of masonry over. Ans. 2090 
tons. 

Fig. 39P- Note. The pull of gravity may be 

thought of as applied at the center of the 
block of masonry, and the problem may be solved by equating torques about the 
point T. If it is desired to find the force which the earth exerts on the pier at T, 
one may proceed as follows: (i) Draw the line of action of the force with which 
the earth pulls on the pier; this is a vertical line passing through the center of the 
pier; let P be the point where this line intersects FC; then TP is the line of 
action of the force exerted on the pier at T, according to Art. 24. (2) Having 




SIMPLE STATICS. 



69 



determined the direction of the force at T we have everything given but the mag- 
nitude of two forces and therefore the problem falls under case c, Art. 20. 

40. Find the force F in Fig. 40^ required to draw a 200-pound 
block up an inclined plane of the dimensions shown, the coefficient 
of friction between block and plane being 0.2. Ans. 95 pounds. 



6 feet 




Fig. 40£. 

Note. The block is in equilibrium under the action of three forces; (1) the 
downward pull of the earth of 200 pounds, (2) the unknown force F (of which the 
direction is known), and (3) the unknown force exerted on the block by the plane. 
This force is in the direction of the line P in Fig. 40^. See Art. 52 for a discussion 
of coefficient of friction. 

41. A wedge of which the shape is indicated in Fig. 4.1 p is 
pushed between two blocks A and B with a force F of 5000 
pounds. The coefficient of 
friction between the wedge 
and the blocks is 0.2. Find 
the components at right an- 
gles to F of the forces with 
which the wedge pushes on 
A and B. Ans. 8676 pounds. 

Note. The lines OP and 0' ' P' 
in Fig. 41^ represent the forces with 
which the wedge pushes against the 
blocks A A and BB respectively. 
The blocks push against the wedge 
with equal and opposite forces. 
Therefore, of the three forces which 
act on the wedge, one is given in 
magnitude and direction, the direc- 
tions of the other two are known, 
and the magnitudes of the other two are to be found 
falls under case (c) in Art. 20. 




Fig. 4ip. 
This problem, therefore, 



7o 



ELEMENTS OF MECHANICS. 



42. The diameters of the two drums D and d in Fig. 42^ are 
4 feet and 1 foot respectively, and the diameters of the pitch 
circles of the two cog-wheels W and w are 5 feet and 10 inches, 
respectively. Find the weight A. Ans. 4800 pounds. 

Note. When the point of contact of 
two cogs on W and w is on the line of the 
centers of the two wheels, the point of con- 
tact is at certain distances r and R respect- 
ively from the centers, where r and R are 
the radii of what are called the pitch circles 
of the two cog-wheels. The cog-wheels may- 
be thought of as-friction drums of which the 
diameters are equal to the diameters of the 
pitch circles of the respective cog-wheels. 
What force balances the torque action of 
the 200-pound weight on the body Dw, what exerts this force, upon what is 
the force exerted, what is the direction of the force, and at what distance from 
the center of w is the line of action of this force? What force balances the torque 
action of the weight A on the drum d, what exerts this force, upon what is the 
force exerted, what is the direction of the force, and how far is the line of action 
of the force from the center of d? 

43. A uniform stick 6 feet long and weighing 10 pounds has 
three weights hung upon it as shown in Fig. 43 £. Find the 




Fig. 42p. 



distance x from the end of the 
stick to the point where the 
single force F must be applied 
to produce equilibrium. Ans. 
3.18 feet. 

44. The steam in an engine 
cylinder pushes on the piston 
with a force of 12,000 pounds- 
weight. The positions and 
lengths of connecting rod and 
crank are shown in Fig. 44/?. 



5% feet 

...x_f e J?L ^ 

3 feet ^ 



1 foot 



10 pounds 

m 

15 pounds |j| 

20 pounds 

Fig. 43P- 



25 pounds 



Find the force with which the cross-head pushes sidewise against 
the guide, the thrust of the connecting rod, and the torque in 
pound-feet exerted on the crank-shaft, neglecting friction through- 
out. Ans. Side-thrust 1257 pounds; thrust in connecting rod 
12065 pounds; torque 5000 pound-feet. 



SIMPLE STATICS. 



71 



45. A ladder 16 feet long and weighing 100 pounds has its 
center of gravity 7 feet from its lower end which stands on a 
floor at a distance of 4 feet from a vertical wall against which 



11 



12000 



pounds 

;;;, ■■.■■■■ v., v;;,;„j si: 




Fig. 44P. 

the ladder rests, as shown in Fig. 45^. Assuming the force a 
with which the wall pushes on the ladder to be horizontal, find 
the magnitude of a and the direction and magnitude of the 
force with which the ladder pushes against the floor. Ans. 
Force a 11.2 pounds; 84 with floor; 100.6 pounds. 

46. An elevator car with its load weighs 1500 pounds and the 
center of gravity of the whole is one foot off center, as shown in 
Fig. 4\6p. The coefficient of friction against the guides is 0.1. 



guide 




wall 



guide 










B 


1 

: c 

ft 

\Z foot 










7f 


set 



6 feet 



Fig. 45£- 



Fig. 46P. 



Find the tension in the cable necessary to draw the car upwards 
at constant speed. Ans. 1550 pounds. 

Note. This problem is solved in a manner exactly similar to the problem of the 
table drawer which is discussed on page 58. 



72 



ELEMENTS OF MECHANICS. 



47. Given a tackle block arranged as shown in Fig. 47^?. Find 
the weight W which can be lifted by a force F equal to 150 
pounds- weight, neglecting friction. Ans. 600 pounds. 

Note. The simplest argument of this problem is as follows: The 
tension of the rope is everywhere equal to 150 pounds if friction 
is negligible. Therefore, the four strands of rope which lead to the 
lower block exert a total lifting force of 600 pounds. 

This problem can be handled by means of the principle of vir- 
tual work. Let d be the distance through which the force F has 
pulled the rope. Then d/4 is the distance that the weight W has 
been lifted, so that Fd is the work done by the force F and PFd/4 
is the work which has been expended in lifting the weight. There- 
fore, ignoring friction, we have Wd/4 = Fd, or W = 4F. 

48. The conical end of a vertical shaft rests in a 
conical seat. The total weight of the shaft and the 
attached wheel is 2000 pounds, and the total area of 
contact surface between the conical end of the shaft 
and the conical seat is 10 square inches. The semi- 
angle of the cone is 20 . Find the pressure p (force 
per unit area) between the conical end of the shaft and the 
conical seat, assuming the force to be everywhere at right an- 
gles to the surface of contact, and assuming p to have the same 
value over the entire surface. Ans. 585 pounds per square inch. 

Note. Problems in statics in three dimen- 
sions (that is, where all the forces under con- 
sideration are not in one plane) are usually 
quite complicated; fortunately such problems 
are not frequently met with in engineering 
practice. 

To solve this problem, consider the force p, 
which acts upon each unit area of the conical 
surface of the end of the shaft. The upward 
component of this force is p sin 20 , and each 
unit of area of the conical end of the shaft con- 
tributes this same amount of upward force, so 
that the total upward force is 10 times p sin 20 . 



Fig. 47£. 




40 feet 



10 feet 

Fig. 49P- 



49. The beam in problem 36 consists of two legs A and B as 
shown in Fig. 49^, and the edgewise view of this double beam 
is as shown in Fig. ^6p. Find the compression in each of the 



SIMPLE STATICS. 



73 



two legs A and B, using the data of problem 36. Ans. 1725 
pounds. 

Note. This is a problem in statics in three dimensions, but it can be easily 
reduced to two problems each in two dimensions. The answer to problem 36 gives 
the downward force F in Fig. 49P in the plane of the two legs A and B, and the 
compression in each of the legs is then easily found by considering the three forces 
which act upon a particle at P in Fig. 49P, namely, the force F and the thrusts of 
the two beams A and B. 

50. Fig. $op represents a top view of a telegraph pole P at a 
point where the telegraph line turns a right angle, the pole P be- 
ing guyed by a wire which is inclined at an angle of 45 ° to the 
horizontal. The horizontal component of the tension of each 
telegraph wire is 250 pounds. Find the tension in the guy wire. 
Ans. 500 pounds. 

Note. This is a problem in statics in three dimensions, but it can be easily 
reduced to two problems each in two dimensions. Find the horizontal component 
of the tension of the guy wire by considering that it must be equal and opposite 



telegraph wire 



guy wire 




telegraph 
wire 



Fig. 5o£. 



3 feet ^lf??§ 



armature 



[Ipulley 



\ 



Fig. Sip. 



to the resultant of the horizontal pulls of the two telegraph wires. Then imagine 
a vertical plane drawn through the guy wire. The horizontal component of the 
tension of the guy wire may be represented by a horizontal line in this plane, and 
the actual resultant tension of the guy wire will be represented by a line inclined 
45° downwards. 

51. Fig. 51^ represents the armature shaft of an electric motor, 
the motor bearings being three feet apart, and the center of the 
pulley being one foot distant from the center of the adjoining 
bearing. The pull of the belt is a horizontal force B of 120 



74 ELEMENTS OF MECHANICS. 

pounds, and the weight of the armature is a downward force of 
150 pounds. Find the direction and magnitude of the force with 
which each bearing pushes against the armature shaft. Ans. 
At bearing a 177 pounds, 25 above horizontal. At bearing b 
85 pounds, 62 above horizontal. 

Note. This is a problem in statics in three dimensions, which can be easily 
reduced to four problems, each in two dimensions. Consider the force B, and 
find the two horizontal forces which must be exerted upon the shaft by the bearings 
to balance this force; then find the vertical forces with which the bearings must 
act upon the shaft to balance the downward pull of the earth on the armature; 
and then find the resultant of the vertical and horizontal forces exerted by each 
bearing. ~ • 



CHAPTER IV. 

DYNAMICS.* TRANSLATORY MOTION. 

28. Force and its effects. When one pushes or pulls on an 
object one is said to exert a force on the object, f Thus the pull 
of a horse on a wagon is called a force. It does not, however, 
require an active agent | like a horse or an engine to exert a force. 
Thus a weight lying on a table exerts a downward force on the 
table, a string stretched between two pegs exerts a force on each 
peg. 

The effects of a force are extremely varied. Thus to pull on 
a body may set the body in motion; to pull on a rubber band 
stretches it; to place a load on a long beam bends or breaks it; 
a coin is heated when force is exerted upon it in rubbing it on a 
board; when steam is compressed it is condensed into water; 
when ice is compressed a portion of the ice melts; etc. Indeed 
nearly every physical phenomenon involves force action of one 
kind or another. 

A force can be measured only in terms of its effects, and the 
effect which can be most easily used for the measurement of 
force is the effect the force has in distorting a body, as, for ex- 

*One of the best treatises on Mechanics for the student is An Elementary 
Treatise on Theoretical Mechanics, by Alexander Ziwet, The Macmillan Company, 
1893. In keeping this work within reasonable bounds the author has excluded 
the more advanced parts of the subject. The book, however, gives references for 
the use of those who may desire to pursue the subject further. 

fThis matter is discussed very briefly on page 5. There has been a great deal 
of discussion, ever since the time of Sir Isaac Newton, concerning the nature of 
force. Perhaps the most significant discussion of this matter is that which is given 
in Appendix B (pages 268-288) of Aether and Matter by Sir Joseph Larmor, Cam- 
bridge University Press, 1909. This discussion of Larmor's is rather difficult to 
follow. A very good simple discussion of the subject may be found in Ernst 
Mach's Mechanics (translated by Thomas J. McCormack), Open Court Publishing 
Company, Chicago, 1907. Every serious student of mechanics should read this 
book of Mach's. 

tSee Art. 53 for a discussion of active and inactive forces. 

75 



y6 ELEMENTS OF MECHANICS. 

ample, in stretching a helical spring. The simplest effect of a 
force, however, is the change which an unblanced force produces in 
the velocity of a body; this effect is the simplest because it is 
independent of the nature of the body.* This effect is now 
universally adopted as the effect in terms of which force is funda- 
mentally measured. 

The study of the effects of unbalanced forces in modifying the 
motion of bodies constitutes the science of dynamics. 

29. Types of motion and types of force action, f Motion, as 
it occurs in nature, is infinite in variety, but there are certain 
simple types of motion such as the forward motion of a boat or 
the rotatory motion of a wheel, and the discussion of these 
simple types of motion constitutes the science of mechanics. 
A body is said to perform translatory motion when every line in 
the moving body remains unchanged in direction. Thus a car 
moving along a straight track performs translatory motion. A 
body is said to perform rotatory motion when a certain line in 
the moving body remains fixed in position. This line is called the 
axis of rotation. Thus the flywheel of an engine performs 
rotatory motion. 

A body may perform various types of motion simultaneously. 
It is better, however, to study each type of motion by itself, 
and some help is afforded towards the keeping of the types of 
motion clearly separated in one's mind by conceiving of ideal 
bodies as follows: 

A material particle is an ideal body so small that the only 
sensible motion of which it is capable is translatory motion. 
The term material particle is used merely to direct one's attention 
to translatory motion, and any body whatever which performs 
translatory motion may be thought of as a particle if one wishes 
to think in such terms. 

A rigid body is an ideal body which cannot alter its shape and 

*This matter is discussed at some length on pages 6-n. 

fOther types of motion and of force action are discussed later. Thus a variety 
of force actions are discussed in the chapter on elasticity, and the ideal simple motion 
of flow of a fluid is discussed in the chapter on hydraulics. 



DYNAMICS. TRANSLATORY MOTION. 77 

which is capable, therefore, of performing only translatory motion 
and rotatory motion. The term rigid body is used merely to 
exclude the idea of change of shape in the discussion of rotatory 
motion. 

The use of ideal bodies in the development of mechanics may 
seem to be objectionable, but it is necessary to discuss one thing 
at a time and it is even more necessary to ignore the interminable 
array of minute effects which always accompany every physical 
phenomenon ; an attempt to consider these minute details would 
complicate every engineering problem beyond the possibility of 
a practical solution. Thus for most practical purposes one may 
think of the motion of a railway car along a straight level track 
as simple translatory motion, whereas the actual motion involves 
the swaying and vibration of the car and the rattling of every 
loose part, it involves a complicated phenomenon of motion which 
is called journal friction, and it involves the yielding of the track 
and a whirling, eddying motion of the air, it is, in fact, infinitely 
complicated; but the railway engineer who, for example, is con- 
cerned with the design of a locomotive of adequate power sums 
up all of these effects in a rough estimate of the total frictional 
drag which the locomotive has to overcome. 

Types of force action. To each type of motion there is a cor- 
responding type of force action. Thus a force action which 
tends to produce translatory motion only is called a linear force, 
and that action of a force which tends to produce rotatory motion 
is called turning force or torque. 

30. Center of mass. The simplest case of translatory motion 
is the motion of a body along a straight path, either with constant 
or varying velocity, as exemplified by the motion of a car along 
a straight track or the motion of a ship on a straight course. 
The most general case of translatory motion, however, is that 
in which a given point of the body describes any path whatever 
in any way whatever, but where every line in the body remains 
unchanged in direction as indicated in Fig. 25. 



7% ELEMENTS OF MECHANICS. 

It is important to understand that there is a certain point in a 
body at which a single force must be applied to produce trans- 
latory motion without rotation; and it is important to under- 

jenteronnas^ Stand that when a b ° d y d ° eS 

W^^^^^^^^^^^E^^ perform translatory motion 

( without rotation, the body may 

, :>. v ^-^_2L-£!i. n be thought of as being con- 

L -T- 1 centrated at this point. This 

"**N point is called center of mass 

Fig 2S> * of the body. Grasp a long 

slim stick at its middle and 
move it up and down and to and fro in any way but without 
changing the direction of the stick; it seems, with the eyes closed, 
as if the stick were a heavy body concentrated in the hand, or, 
in other words, the stick behaves as if it were all concentrated 
at its middle point which is its center of mass.* 

31. Displacement, velocity and acceleration, f When a par- 
ticle moves from one position to another it is said to be displaced, 
and the distance (and direction) from the initial to the final 
position of the particle is called its displacement. 

The displacement of a particle divided by the time during 
which the displacement takes place is called the average velocity 
of the particle during that time. A particle may move in any 
way whatever in making a given displacement and it is therefore 
evident that the actual velocity of the particle at successive in- 
stants during the displacement may be very different from the 
average velocity; but if the interval of time is extremely short 
(and, of course, the displacement small) then all irregularities 

*Every student should perform this experiment for himself; no amount of ar- 
gument will give him so vivid an idea of the meaning of the above statements 
concerning center of mass, namely (a) that a single force must be applied at the 
center of mass of a body to produce translatory motion and (b) that a body may 
be thought of as concentrated at its center of mass in so far as translatory motion 
only is concerned. Center of mass is discussed in Arts. 48, 49 and 50. 

tThe discussion of the time variation of velocity which is given in Art. 18 should 
be reviewed at this point. 



DYNAMICS. TRANSLATORY MOTION. 79 

vanish in accordance with the principle of continuity as stated 
in Art. 17. Therefore, the average velocity of a particle during a 
very short interval of time is the actual velocity of the particle at 
a given instant. 

When the velocity of a particle is changing, the actual change 
during a given interval of time divided by the interval is called 
the average acceleration of the particle during the interval ; and if 
the interval is very short, the acceleration so defined is the actual 
acceleration of the particle at a given instant. 

The actual velocity of a particle at a given instant is of course 
never determined by an attempt to observe the displacement 
during a very short interval of time, and the actual acceleration 
of a particle at a given instant is of course never determined by 
an attempt to observe the change of velocity during a very short 
interval of time. The above definitions do not refer to observa- 
tion but to thinking, as explained in Art. 17. 

32. Newton's laws of motion. The laws of motion are dis- 
cussed in detail on pages 2-1 1. A usual form of statement of 
these laws is as follows : 

I. All bodies persevere in a state of rest, or in a state of uni- 
form motion in a straight line, except in so far as they are made to 
change that state by the action of an unbalanced force. 

II. (a) The acceleration of a particle is parallel and propor- 
tional to the unbalanced force acting on the particle. 

(b) The acceleration which is produced by a given unbalanced 
force is inversely proportional to the mass of the particle. 

III. Action is equal to reaction and in a contrary direction. 

A clear understanding of the laws of motion is of utmost im- 
portance, and the following modified statements may serve to 
make their meaning clear. 

(1) The first law describes the behavior of a particle upon which 
no unbalanced force acts. The behavior is simply that the velocity 
of the particle does not change, and, conversely, if a body move 
at uniform velocity in a straight line, the forces which act upon it 
are balanced. 



8o ELEMENTS OF MECHANICS. 

(2) The second law describes the behavior of a particle when 
acted upon by an unbalanced force. The behavior is simply that 
the particle gains velocity in the direction of the force at a rate which 
is (a) proportional to the force and (b) inversely proportional to 
the mass of the particle. 

When Newton made this statement he, of course, had it in mind 
that a force was measured by some effect other than acceleration, 
otherwise he could not have affirmed, except as a definition, that 
the acceleration which a force produces is proportional to the 
force. The production of acceleration is now adopted as the 
effect by which forces are measured. 

The second law may be further paraphrased as follows: (a) 
The amount of velocity gained by a given particle in a given 
interval of time is proportional to the unbalanced force acting on 
the particle, and the gained velocity is parallel to the force. 
(b) The amount of velocity produced by a given unbalanced force 
in a given interval of time is inversely proportional to the mass 
of the particle upon which the force acts, and the gained velocity 
is, of course, parallel to the force. 

A common form of statement of Newton's second law is that 
the effect of a force is the same whether it acts alone or in conjunction 
with other forces, meaning, of course, the accelerating effect.* 
To see that this statement of the second law is equivalent to the 
statements previously given, let it be understood that any effect 
which is proportional to a cause may be divided into parts and each 
part assigned as the effect of a corresponding part of the cause. Thus 
if the result of the labor of a number of men is proportional to 
the number of men, then it is justifiable from physical considera- 
tions to give each man an equal share of the profits; but if the 
result is not proportional to the number of men, it is not justifiable 
physically to give to each man an equal share of the profits. The 
principle of dividing cause and effect into parts which correspond 

*Every schoolboy knows that, in general, the effect of a force is not the same 
whether it acts alone or in conjunction with other forces; two boys on thin ice may 
produce an effect which is absolutely different from anything which can be produced 
by the weight of one boy alone. 



DYNAMICS. TRANSLATORY MOTION. 8 1 

each to each when cause and effect are proportional, is called the 
principle of superposition and it runs through the whole science 
of physics and chemistry. 

An example of Newton's second law is the familiar fact that 
two bricks fall at the same increasing velocity as one brick; the 
total force with which the earth pulls on the two bricks produces 
velocity at the same rate as the single force with which the earth 
pulls on one brick. That is, to double the mass of a body (two 
bricks instead of one) and at the same time to double the force 
acting on the body (pull of earth on two bricks instead of pull 
of earth on one brick) leaves the acceleration unchanged. 

(3) The third law expresses the fact that a force is always due 
to the mutual action of two bodies, that this mutual action always 
consists of a pair of equal and opposite forces, and that one of 
these forces acts on body number one and the other upon body 
number two. The mutual force action between two bodies is 
called action in its effect upon the body which is being studied 
and reaction in its effect upon the body which is not being par- 
ticularly studied, in the same way that a trade is called buying as 
it effects one person or selling as it affects the other person. 

Inertia. That property of a particle by virtue of which it per- 
severes in a state of rest or in a state of uniform motion in a 
straight line when not acted upon by an unbalanced force is 
called inertia; and the word inertia is generally extended in its 
meaning to include, not only this passive property, but also the 
idea of reluctance to gain velocity. Thus a given unbalanced force 
would have to act for a longer time on a body of large mass than 
upon a body of small mass to produce a given velocity, that is, 
the body of large mass has the greater reluctance to gain velocity, 
or a greater inertia in the extended sense of that term. 

33. Dynamic units of force. Formulation of the second law 
of motion. Having agreed to measure a force in terms of its 
effect in changing the velocity of a particle, we may choose as 
our unit of force that force which, acting as an unbalanced force 

7 



82 ELEMENTS OF MECHANICS. 

on unit mass, will produce unit velocity in unit time (unit ac- 
celeration). Thus the dyne is that force which, acting for one 
second as an unbalanced force on a one-gram particle, will pro- 
duce a velocity of one centimeter per second (an acceleration of 
one centimeter per second per second) ; and the poundal is that 
force which, acting for one second as an unbalanced force on a 
one-pound particle, will produce a velocity of one foot per second 
(an acceleration of one foot per second per second). The dyne 
is the c. g. s. unit of force and it is much used; the poundal is 
seldom used. 

Having adopted as our unit of force that force which will pro- 
duce unit acceleration of unit mass, it is evident, from Newton's 
second law, that F units of force will produce F units of accelera- 
tion of a particle of unit mass, or Fjm units of acceleration of m 
units of mass; that is, Fjm is equal to the acceleration a which 
the force F produces or Fjm = a, whence 

F = ma (i) 

in which F is the value of an unbalanced force in dynes (or 
poundals), m is the mass of a particle in grams (or pounds), and 
a is the acceleration in centimeters per second per second (or feet 
per second per second). 

Gravitational units of force. Weight. The force with which 
the earth pulls on a body is called the weight of the body. The 
precise meaning of the terms mass and weight, as these terms are 
used in scientific writing, is explained in Art. 5. The force with 
which the earth pulls on a one-pound mass (on a one-pound body) 
is called a pound-of -force or a pound-pull* The pound-of-force 
is used as the unit of force in most practical work. Thus we 
speak of 5000 pounds-of -force, meaning a force equal to that with 
which the earth pulls on a 5000-pound body. 

The pull of the earth on a one-pound body is about one-quarter 
of one per cent, greater at 6o° north latitude than it is at 30 north 
latitude, and it is about one-sixteenth of one per cent, less at 

*The gram-oj "-force or the gram-pull is defined in a similar manner. 



DYNAMICS. TRANSLATORY MOTION. 83 

15,000 feet above sea level than it is at sea level.* This slight 
variation in the value of the pound-of-force at different places 
on the earth is of no consequence, however, in those cases where 
the pound-of-force is used as a unit. Thus the tensile strength 
of a given grade of steel, repeatedly measured under conditions 
as nearly alike as it is possible to make them, will vary from, 
say, 100,000 pounds-of-force per square inch to 105,000 pounds- 
of-force per square inch, that is, the tensile strength of a given 
grade of steel is inherently indefinite (like the length of an angle- 
worm!), and a variation of a few tenths of one per cent, in the 
value of the unit of force is of no consequence whatever. 

Relation between dynamic and gravitational units of force. 

A force of one dyne will produce an acceleration of one centi- 
meter per second per second when it acts as an unbalanced force 
on a one-gram body; the pull of the earth on a one-gram body 
produces an acceleration of about 980 centimeters per second 
per second; therefore the pull of the earth on a one-gram body 
is equal to about 980 dynes. 

A force of one poundal will produce an acceleration of one 
foot per second per second when it acts as an unbalanced force 
on a one-pound body; the pull of the earth on a one-pound body 
produces an acceleration of about 32.2 feet per second per second; 
therefore one pound-of-force is approximately equal to 32.2 poun- 
dals. 

Let W be the force, expressed in dynamic units, with which 
the earth pulls on a body, then according to equation (1), we have 

W = mg (2) 

in which W is the weight of the body in dynes (or poundals), 
m is the mass of the body in grams (or pounds), and g is the 
acceleration due to gravity expressed in centimeters per second 
per second (or in feet per second per second). 

*The acceleration of gravity at 45 north latitude and at sea level is 980.6 cen- 
timeters per second per second, and the acceleration of gravity at latitude § and 
at an elevation of H meters above sea level is: 

g = 980.6 (1 — 0.0026 cos 2<t> — o. 0000002 H) cm. /sec 2 . 



84 ELEMENTS OF MECHANICS. 

Form of equation (1) when F is expressed in pounds-of -force, 
mass in pounds, and acceleration in feet per second per second. 

One pound-of-force will produce an acceleration of about 32.2 
feet per second per second when it acts as an unbalanced force 
on a one-pound body, or an acceleration of 32.2/ra feet per second 
per second when it acts on a body of which the mass is m pounds, 
and F pounds-of-force will produce F times as much acceleration, 
or an acceleration of (32.2 F)/m when it acts upon a body of 
which the mass is m pounds. That is to say, the acceleration a 
which is produced by F pounds-of-force acting upon a mass of m 
pounds is 

32. 2F 
a = (3) 

whence 

F = • ma (4.) 

32.2 v ^ y 

or, using g for the acceleration of gravity in feet per second per 
second, we have 

F=~-ma (5) 

The slug as a unit of mass. Many engineers prefer to write equation (5) in 
the form 

-(f)" 

and in order to bring this equation into the form of equation (1) these engineers 
speak of ml g as the mass of the body. It must be kept in mind, however, that the 
mass of a body ivhen so expressed is not expressed in pounds. In buying sugar or 
coal by the pound the word pound is used in its legitimate sense as a unit of mass; 
and to express the mass of a body in terms of the ratio ml g is to adopt 32.2 pounds 
as the unit of mass. This unit of mass is sometimes called the "gee-pound" or 
the "slug." Throughout this text mass will be expressed in pounds, force in pounds. 
pull and acceleration in feet per second per second, except where c. g. s. units are 
used, and equation (5) will always be used when English units are employed. 

34. Measurement of force, (a) By the kinetic method. The 
force (unbalanced) acting on a body may be calculated by equa- 
tion (1), the mass of the body being known and the acceleration 
being determined by observation. This method for measuring 



DYNAMICS. TRANSLATORY MOTION. 



85 



force cannot be realized in its simplicity, but it forms the basis 
of many physical measurements. 

(b) By the counter-poise method. The strengths of materials 
are nearly always determined by applying, as the breaking force, 
the weight of a body or bodies of known mass, multiplied in a 
known ratio by a system of levers. The machine for carrying 
out such a test is called a testing machine and it is similar in 
many respects to the ordinary platform balance-scale. 

(c) By means of the spring scale. The spring scale is an ar- 
rangement in which an applied force stretches a spring and 
moves a pointer over a divided scale. The movement of the 
pointer is proportional to the force, and, the movement for a 
known force being observed, the scale can be divided so as to 




Fig. 26. 

read the value of any force directly. The use of the spring- 
scale is exemplified in the measurement of the draw-bar pull of 
a locomotive. Figure 26 shows a scale designed for this purpose. 
The blocks A and B are rigidly fixed to the "dynamometer car" 
and the link H couples with the locomotive. A pull on H moves 
the cross bar C and compresses the springs, and a push on H 
moves the cross bar D and compresses the springs. The relative 
motion of E and B actuates a pointer which plays over a divided 
scale. 



86 



ELEMENTS OF MECHANICS. 



UNIFORMLY ACCELERATED TRANSLATORY MOTION. 

35. Falling bodies. When a constant* unbalanced force acts 
upon a particle, the particle gains velocity at a constant rate. 
Such a particle is said to perform uniformly accelerated motion. 
A body falling freely under the action of the constant pull of the 
earth is, in so far as the friction of the air is negligible, an example 
of uniformly accelerated motion. 

All bodies when falling freely gain velocity at the same rate, air 
friction being negligible. Thus two bricks together fall at exactly 
the same increasing speed as one brick alone, The doubled pull 
of the earth on the two bricks produces the same acceleration as 
the single pull of the earth on one brick. Doubling the force 
and doubling the mass leaves the acceleration unaltered. 

Consider a particle which gains velocity at a constant rate of 
g centimeters per second per second, a falling body for example. 
The velocity gained in t seconds is 

v = gt (i) 

Let v x be the initial velocity of the particle. Then v 1 + gt is 
its velocity after t seconds, and its average f velocity during the t 
seconds is %[v Y + (y l + gt)] or v 1 + %gt\ and the distance d 

♦Constant in magnitude and unchanging in direction. 

fLet the constantly increasing velocity of a falling body be represented by the 
ordinates of a curve of which the abscissas represent elapsed times. The "curve*. 



axis of velocity 




so plotted will be a straight line A B, Fig. 27, and the average ordinate of any por- 
tion AB of this line is equal to /4(v x +z> 2 ). 



DYNAMICS. TRANSLATORS MOTION. 



87 



fallen by the particle during the t seconds is equal to the product 
of the average velocity by the time /. That is, 

d = Vjt.+ y 2 gt 2 (ii) 

If v x is zero, equation (ii) becomes 

d = y 2g t 2 (ih) 

Eliminating t between equations (i) and (iii) , we have 

v = V / 2gd (iv) 

which expresses the velocity of a body after it has fallen a dis- 
tance d (initial velocity zero) . 

This discussion of falling bodies exemplifies the method of 
integral calculus. See footnote on page 38. 

36. Projectiles. When the initial velocity v x of a body is 
zero or when it is vertical, we have the ordinary case of a falling 
body, and equation (ii) of Art. 35 can be solved by simple arith- 
metic, the only complication being that v l is to be considered 
negative when it is upwards. When the initial velocity v x is not 
vertical, as in the case of a tossed ball, the falling body is called 
a projectile. In this case the entire argument of Art. 35 holds 





Fig. 28a. 



Fig, 28&. 



good but geometric addition must be substituted for arithmetic 
addition. Thus the average velocity of a projectile during t 
seconds is equal to the geometric sum, v x + }4gt, as shown in Fig. 
28a, and after t seconds the projectile is on the line OB at a dis- 
tance trom equal to t times the numerical value of the average 



88 



ELEMENTS OF MECHANICS. 



velocity. Or, one may find the position of the ball after t sec- 
onds on the basis of equation (ii) , considering that v x t is a distance 
in the direction of v v that }4gt 2 is a distance vertically downwards, 
and that the sum v x t + yigtr is a geometric sum as shown in 
Fig. 286. 

The orbit of a projectile is a parabola. This may be shown 
by choosing the x-axis of reference parallel to v x and the ^-axis 
vertically downwards. Then x = v x t and y = }4gt 2 , whence, by 
eliminating t we have the equation of the parabola. 

The hodograph to the orbit of a projectile is a vertical straight 
line. Draw the line OP, Fig. 29, representing the velocity of a 

projectile at a given distant, 
then, after t seconds, the vertical 
velocity gt will be gained, and 
the total velocity will be repre- 
sented by OP' . Therefore, if we 
imagine the line OP to change so 
as to become OP' after t seconds 
and thus represent the changing 
velocity at each instant, then 
the end P will move vertically 
downwards at a constant velocity. 

Range of a projectile. The horizontal distance reached by a 
projectile when it comes to the level of the gun on its downward 
flight is called the range of the projectile. The range of a pro- 
jectile, ignoring the effects of air friction, is given by the equation 

2v 2 sin 6 cos 6 




Fig. 29. 



I = 



0) 



in which v l is the initial velocity of the projectile, 9 is the angle 
between the direction of v 1 and a horizontal line, and g is the 
acceleration of gravity. This expression for I may be easily de- 
rived with the help of the relations shown in Fig. 30, namely, 



I = v,t cos 6 



and 



%gt 2 = v x t sin B 
whence, eliminating t, we have equation (i). 



DYNAMICS. TRANSLATORY MOTION. 



89 



37. Effect of air resistance on the motion of a projectile. 

.Bodies which are projected through the air do not have a con- 
stant downward acceleration, because of the resistance which the 
air offers to their motion, and therefore the simple theory of pro- 
jectiles above outlined is not applicable in practice. The limita- 
tions of this simple theory may be stated in a general way as 
follows : 

(a) In the first place the above simple theory is not limited to 
the motion of an ideal particle. The pull of the earth upon a 
projectile tends only to produce translatory motion and the center' 

id 




~a¥L.~ 



of mass of the body describes in every case a smooth parabolic 
curve in accordance with the discussion of Art. 36, air friction 
being ignored. Thus, if an iron bar is thrown through the air, the 
center of mass of the bar describes a smooth parabolic orbit ; or if 
the bar is projected by hitting it a sharp blow with a hammer, 
the center of mass of the bar describes a smooth parabolic orbit. 
This illustrates a very important extension of the idea of a ma- 
terial particle, namely, we may call any body a material particle, 
whatever the character of its motion may be, the idea being to 
direct one's attention solely to that part of the motion of the 
body which is translatory. 

(6) In the case of a heavy body moving slowly, for example, 
an iron ball tossed from the hand, the resisting force of the air is 



90 



ELEMENTS OF MECHANICS. 



very small compared with the weight of the body, and the motion 
of the body approximates very closely indeed to the ideal motion 
discussed in Art. 36. 

(c) In the case of a light body, or in the case of a heavy body 
projected at high velocity, the resisting force of the air may be 

orbit in vacuum (parabola) range 28 miles 




K —range 16 miles 

Fig. 31. 

very large, so that the motion of such a body differs widely from 
the ideal motion described in Art. 36. Thus, Fig. 31 shows the 
actual orbit of the heavy projectile from a modern high power 
gun, and the dotted line shows what the orbit would be in a 
vacuum. 

(d) The air friction on a rotating projectile generally gives rise 
to a force which pushes the projectile side wise. This side force 




is the cause of the curiously curved orbit of a "split-shot" 
tennis ball, and of a base ball pitched by an expert pitcher. The 



DYNAMICS. TRANSLATORY MOTION. 9 1 

curved arrows, Fig. 32, show the direction of rotation of a base 
ball, the arrow M shows the direction of its translatory motion, 
the arrow F shows the side force above mentioned, and the dotted 
curve shows the curved orbit.* 

TRANSLATORY MOTION IN A CIRCLE. 

38. Velocity and acceleration of a particle moving steadily 
in a circular orbit. Consider a particle which makes, steadily, 
n revolutions per second in a circular orbit of radius r. The 
circumference of the orbit is 2irr, and, inasmuch as the particle 
traverses the circumference n times per second, its velocity v is 

v = 2irrn (6) 

The magnitude, or numerical value, of the velocity v is con- 
stant; but its direction is changing continuously, this continual 
change of direction of v involves acceleration, and the state of 
affairs at each instant during the steady motion of a particle in a 
circular orbit is most clearly shown by the use of the idea of the 
hodograph as explained in Art. 18. It is instructive, however, 
to discuss the motion of a particle in a circular orbit without ex- 
plicit reference to the hodograph, as follows: 

To determine the acceleration of a particle which is moving 
steadily in a circular orbit, it is necessary to consider the change 
of velocity during a very short interval of time. The circle, 
Fig. 33, represents the orbit of the particle, and at a given instant 
the particle is at P. At this instant the velocity v l of the particle 
is at right angles to PO and it is represented by the line 0' P r 
which is drawn from the fixed point 0' . After the small lapse of 
time At, the particle will have moved a distance v • At to the point 
Q, and its velocity will be v 2 , which is represented by the line O'Q' . 
The change of velocity Av is evidently parallel to PO (or to 
QO, for it must be remembered that the time interval At is in- 
finitely small), and, since the triangles OPQ and 0' P'Q' are 
similar, we have 

♦This matter is discussed again in the chapter on hydraulics. 



92 ELEMENTS OF MECHANICS. 

Av v At 



v 



CO 



in which v is written for the common numerical value of v 1 and 
v 2 , and v-At is the length of the infinitesimal arc PQ which is 




or 



Fig. 33- 

traversed by the particle during the time interval At. From 
equation (i) we have 



Av v 2 
At ~ 7 



(ii) 



but, the change of velocity Av divided by the time interval At 
during which the change takes place is the acceleration, so that, 
writing a for Av/At, equation (ii) becomes 

The direction of a is, of course, parallel to Av, and Azj is par- 
allel to PO. Therefore a particle which moves steadily in a 
circular orbit of radius r has a steady acceleration towards the 
center of the circle, and this acceleration is equal to v 2 /r, where 
v is the steady velocity of the particle. 

It is sometimes convenient to have a expressed in terms of r 
and n, thus we may substitute the value of v from equation (6) 
in equation (7) and we have 



DYNAMICS. TRANSLATORY MOTION. 93 

a = 4ir 2 n 2 r (8) 

Force required to constrain a particle to a circular orbit. When 
a piece of metal is tied to a string and twirled in a circular orbit 
the string pulls steadily on the piece of metal, this pull of the 
string is an unbalanced force since no other force* acts on the 
piece of metal, and the value of the force in dynamic units is equal 
to the product of the mass of the particle and its acceleration, 
according to equation (i). Therefore we may substitute the 
value of a from equation (7) or equation (8) in equation (1) 
giving 

F=^ f (9) 

and 

F = 47rVrwt (10) 

where F is the force in dynamic units required to constrain a 
particle of mass m to a circular orbit of radius r, v is the velocity 
of the particle, and n is the number of revolutions per second. 

39. Examples of motion in a circle, (a) A one-pound piece 
of metal twirled five revolutions per second in a circle four feet in 
radius would, according to equation (10), require a force of 3,948 
poundals or about 123 pounds-weight to constrain it to its orbit. 

(b) Each particle of a rotating wheel must be acted upon by an 
unbalanced force to constrain the particle to its circular path. 
If we consider only the rim of the wheel, neglecting the effect of 
the spokes, it is evident that the necessity of the unbalanced radial 
forces gives rise to a state of tension in the rim. The tension in 
a barrel hoop presses each portion of the hoop radially against 
the barrel staves, and the outward push of the staves balances 

*Resistance of the air and force of gravity are here ignored. 

fThese equations express F in dynamic units, dynes or poundals as the case may 

be. If F is to be expressed in pounds-weight these equations become F= mv 2 jr 

32.2 

and F= (4^ 2 n 2 rm), where m is the mass in pounds of the moving particle, v is 

32.2 

its velocity in feet per second, r is the radius of the circle in feet, and n is the number 

of revolutions per second. 



94 



ELEMENTS OF MECHANICS. 



the radial force due to the tension of the hoop; but the tension 
in the rim of a rotating wheel produces an unbalanced radial 
force on each particle of the rim, and this force produces the 
radial acceleration of each part of the rotating hoop. 

(c) The tension of a belt produces a radial force which presses 
the belt radially against the face of the pulley. When the belt 
and pulley are in motion, however, a portion of the belt tension 
produces the radial forces required to constrain the particles of 
the belt to their circular paths ; the portion of the belt tension so 
used is proportional to the square of the velocity of the belt and 
inversely proportional to the radius of the pulley (a = v 2 /r). 
Therefore, belts running at high speeds on small pulleys have a 
troublesome tendency to slip, unless the tension is very great. 

(d) The centrifugal drier which is used in laundries and in 
sugar refineries is a rotating bowl AB, Fig. 34, with perforated 




lib 

MM!" 1 1 



Fig. 34- 




sides, in which the material MM to be dried is placed. The 
action of the centrifugal drier may be clearly understood as fol- 
lows: Consider two solid particles a and b, Fig. 35, with a drop 
of water d clinging to them. Gravity, of course, pulls on the drop 
and the drop adheres to a and b so that the particles are able to 
exert on the drop a force F sufficient to balance gravity. In the 
centrifugal drier, however, the particles would have to exert upon 



DYNAMICS. TRANSLATORY MOTION. 



95 



the drop a force equal to \ir 2 n 2 rm, where r is the radius of the 
circular path described by the particles a and b, m is the mass of 
the drop, and n is the speed of the drier bowl in revolutions per 
second, and this force \-K 2 n 2 rm may be, say, iooo times as great 
as the weight of the drop; but the drop does not have sufficient 
adherence to the particles to enable the particles to hold to it with 
so great a force, and the result is that the drop is not constrained 
to the circular path, but flies off tangentially. The action of the 
centrifugal drier is as if a piece of wet cloth were jerked so quickly 
to one side as to leave the water behind. 

(e) A locomotive on a railway curve describes a circular path 
and an unbalanced horizontal force (equal to mv 2 fr) must push 
the locomotive towards the center of the curve in order that the 




/ / 




Fig. 36. 



locomotive may follow the curve, and, of course, this horizontal 
force must be exerted on the locomotive by the track. It is 
desirable, however, that the total force with which the track 
pushes on the locomotive (which is equal and opposite to the force 
with which the locomotive pushes on the track) shall be perpen- 
dicular to the plane of the track, and, therefore, the outside rail 
is always raised on a railway curve. 



96 ELEMENTS OF MECHANICS. 

Let us consider the proper elevation to be given to the outside 
rail when the velocity v of the locomotive and the radius r of the 
curve are given. The rails are shown at a and b in Fig. 36, F is 
the total force that must act upon the locomotive, and the angle 6 
is the required elevation.* 

The vertical component of F is what sustains the locomotive 
against gravity. Therefore this vertical component is equal to 
mg where m is the mass of the locomotive and g is the accelera- 
tion of gravity. That is : 

F cos = mg K . (i) 

The horizontal component of F is the unbalanced force which 

constrains the locomotive to its circular path. Therefore this 

horizontal component is equal to v 2 m/r according to equation 

(9). That is: 

-^ . v 2 ™* 

F sin 6 = (ii) 

r 

Therefore, dividing equation (ii) by equation (i), member by 
member, we have 

v 2 
tan 6 = — (in) 

rg 

When a locomotive is traveling on a curve it is evident that 
the whole locomotive is rotating about a vertical axis at such an 
angular speed that if the curve were a complete circle the loco- 
motive would make one rotation about a vertical axis every time 
it traversed the circular curve. Therefore a locomotive travel- 
ing on a curve does not perform pure translatory motion; but 
here again is an instance where the translatory motion may be 

*Some of the members of every class in elementary mechanics seem to have the 
idea that the tendency is for the outer wheels of a carriage to rise off the ground 
when the carriage is driven rapidly around a curve. This idea comes from the fact 
that the outer wheels of a carriage must be elevated (by raising the outer part of 
the road) to make the carriage ride around the curve with perfect safety, and a 
bicycle rider leans inwards in order that he may ride around a curve without falling 
over. The leaning over of a bicycle rider as he rounds a curve is not due to the 
mechanical actions involved but to the deliberate control of the rider. 



DYNAMICS. TRANSLATORY MOTION. 97 

considered by itself, for as long as the locomotive is on the curve, 
its rotating motion is constant and introduces no complication. 
When a locomotive suddenly enters a curve from a straight 
portion of track, the rotatory motion of the locomotive about a 
vertical axis must be suddenly established, and in order to estab- 
lish this rotatory motion an excessively great horizontal side- 
force must act on the front wheels of the locomotive ; or in other 





Fig. 37- Fig- 38- 

words, the front wheels of a locomotive must push with excessive 
force against the outer rail when the locomotive enters a curve 
suddenly. This action is called nosing, and it is especially trouble- 
some in the case of a locomotive having a short wheel base. 
When the wheel-base of a locomotive is long, a much smaller 
side force need act upon the front wheels of the locomotive when 
it enters a curve, and therefore the nosing is less troublesome. 
The electric locomotives of the New York, New Haven & Hart- 
ford Railroad when first constructed had very short wheel-bases, 
and considerable trouble was encountered until pilot trucks were 
placed at both ends. 

Fig. 37 shows a straight portion of a track changing abruptly 
to a circular curve at the point a, and Fig. 38 shows the same 
straight portion changing gradually into the same circular curve. 
The slow transition from straight to curved track in Fig. 3 8 con- 
stitutes what is called an easement curve, the object of which is 
to avoid the effects of abrupt entry of locomotive into a curved 
portion of track, as above described. 

40. The rotating hoop. It is pointed out in the above ex- 
amples of circular motion that the radial forces which constrain 
8 



9 8 



ELEMENTS OF MECHANICS. 



the particles of the rim of a rotating wheel to their circular paths 
are (ignoring effect of spokes) due to a state of tension in the 
rim. The tension of the rim is the force T with which any por- 
tion of the rim pulls on a contiguous portion. Let the circles, 
Fig. 39, represent a hoop of a radius r rotating n revolutions 
per second about the axis C, let the mass per unit length of the 
rim of the hoop be m', and let F be the unbalanced force pull- 
ing radially inwards on each unit length of the rim (due to the 
tension in the rim) . Consider a very short portion of the rim of 
length r-A<p, which subtends the angle A<p as shown. The 




Fig. 40. 



unbalanced force pulling the portion r-A<p radially inwards is 
F X r-A<p. Let T 2 be the force with which portion k pulls on 
the given portion r-A<p at the point a, and let T x be the force 
with which the portion j pulls on the given portion r-A<p at the 
point b. The forces T\ and T 2 are equal, numerically, and they 
are tangent to the circle at a and b respectively. Draw the two 
lines 7\ and T 2 , Fig. 40, parallel to T x and T 2 , Fig. 39, and com- 
plete the parallelogram of which T x and T 2 are the sides. The 
diagonal of this parallelogram represents the resultant of T\ 
and T 2 , this resultant is the total unbalanced force (F X r-A<p) 



DYNAMICS. TRANSLATORY MOTION. 99 

which acts on the given portion r • A<p of the rim, and from the 
similar triangles of Fig. 39 and Fig. 40 we have 

r-A<p Fr-A<p 
~r~ = T 

in which T is written for the common numerical value of J\ and 

T 2 . Therefore 

T 

p = - CO 

That is, the unbalanced inward pull on each unit length of a 
hoop is equal to the tension of the hoop divided by its radius. 

Now the mass of unit length of the rim is equal to m! \ and, 
according to equation (10), the force which must be pulling 
radially inwards on unit length of the rim to constrain it to its 
circular path, is equal to ^ir 2 n 2 r times its mass m'. Therefore 
writing \-K 2 r?rm' for F in equation (i) we have 

T = 47rW 2 m' (11) 

in which T is the tension in dynes in a rim r centimeters in radius, 
rotating n revolutions per second, and m! is the mass in grams 
of one centimeter of the rim. If m' is expressed in pounds mass 
per foot of rim, r in feet, n in revolutions per second and T in 
pounds-weight, then 

r / 2 2 2 »\ 

= ( 47r firm) 

32-2 

Example. The rim of a large flywheel has a mass of 250 
pounds per foot, the radius of the wheel is 15 feet, the wheel 
rotates one revolution per second, and the tension of the rim 
(neglecting the effect of the spokes) is 69,350 pounds-weight. 

TRANSLATORY HARMONIC MOTION. 

41. Definition of harmonic motion. Utility of the idea. 

Simple harmonic motion is the projection on a fixed straight line 
of uniform motion in a circle. Consider a point P', Fig. 41, 
moving uniformly around a circle of radius r at a speed of n 



IOO 



ELEMENTS OF MECHANICS. 



revolutions per second, the point P, which is the projection of P' 
on the line CD, performs simple harmonic motion. 

Vibration or cycle. One complete up-and-down movement of 
the point P, Fig. 41, is called a vibration or a cycle. 




Fig. 42. 



Frequency. The number of vibrations, or cycles, per second is 
called the frequency oi the oscillations of the point P; this is, of 
course, equal to the number of revolutions per second of the 
point P'. 

Period. The time required for the particle to complete one 
whole vibration, or cycle, is called the period of the harmonic 
motion. The relation between the frequency n and the period r 
is obviously 

n = - (12) 

T 

Equilibrium position. When the vibrating particle P, Fig. 41, 
is at the point 0, no force acts upon it, as explained below; the 
point is therefore called the equilibrium position of the vibra- 
ting particle. 

Amplitude. The maximum distance from reached by the 
vibrating particle is called the amplitude of its oscillations. This 
amplitude is equal to the radius r of the circle in Fig. 41. 



DYNAMICS. TRANSLATORY MOTION. IOI 

Phase difference. Consider two points P' and Q', Fig. 42, 
both making n revolutions per second around. the circle so that 
the angle 6 is constant. The two oscillating particles P and Q 
are then said to differ in phase and the angle is called their 
phase difference. 

The ideas involved in the peculiar type of motion which is per- 
formed by the particle P, in Fig. 41, are used throughout the 
study of oscillatory motion and wave motion. Thus the prongs 
of a vibrating tuning fork perform simple harmonic motion; the 
motion of a pendulum bob is, approximately, simple harmonic 
motion; when a rod, or a beam, or a bridge oscillates in the 
simplest possible manner, each particle of the rod, or beam, or 
bridge performs simple harmonic motion ; when wave-motion of 
the simplest kind spreads through a body each particle of the 
body performs simple harmonic motion. 

An example of simple harmonic motion in which all of the 
details in Fig. 41 are reproduced, is the motion of the cross-head 
of a steam engine with a long connecting rod. The crank pin 
moves at sensibly uniform speed in a circle, one component, 
only, of this motion is transmitted to the cross-head by the long 
connecting rod, and the cross-head moves to and fro in the 
manner of the point P in Fig. 41. 

42. Acceleration of a particle in harmonic motion. The veloc- 
ity of the point P in Fig. 41 is the vertical component of the 
velocity of the point P' , and the acceleration a of the point P is 
the vertical component of the acceleration a f , therefore, from the 
similar triangles P'OP and P'cd of Fig. 41 we have 

a x 



where x is the distance OP, and, since a' = /\.T 2 n 2 r, according 
to equation (8), we have 

a = — \-K 2 n 2 x. (13) 

The minus sign is introduced for the reason that a is downwards 



102 ELEMENTS OF MECHANICS. 

(negative) when x is upwards, (positive) , and this sign has noth- 
ing to do with the numerical relations under discussion. 

The force which must act on the particle P, Fig. 41, to cause 
it to move in the prescribed manner, is at each instant equal to 
ma, according to equation (1), therefore 

F = — 4.ir 2 n 2 mx (14) 

in which m is the mass of the oscillating particle, x is the distance 
of the particle from its equilibrium position at a given instant, n 
is the frequency of the oscillations, and F is the force which must 
act on the particle at the given instant. * . 

The quantities n and m in equation (14) are constant. There- 
fore equation (14) indicates that the force F which must act at 
each instant on a particle in harmonic motion is proportional to the 
distance x of the particle from its equilibrium position, that is, we 
may write 

F = - kx (15) 

where 

k = 471- VW (16) 

or using i/r for n, where r is the period of one complete oscilla- 
tion, we have 

k = !L i- (17) 

T 

43. Examples of the application of equations (15), (16) and 
(17). (a) Application to a weight attached to the end of a flat 
spring. A weight of mass m is fixed to one end of a flat steel 
spring S, the other end of which is clamped in a vise as shown 
in Fig. 43. If the weight M t is pushed to one side through a dis- 
tance x, the spring exerts a force F which urges the weight back 
towards its equilibrium position and this force is proportional to 
x. Therefore, we may write 

F = - kx (i) 

in which k is a constant, the value of which may be determined 
by observing the force required to hold the weight at a measured 
distance x from its equilibrium position. 



DYNAMICS. TRANSLATORY MOTION. 



103 



Now since equation (i) is identical to equation (15) it is 
evident that the weight, once started, will perform simple har- 
monic motion. 

(b) Application to the simple pendulum. The simple pendulum 
is an ideal pendulum consisting of a particle P, Fig. 44, suspended 



8 




Fig. 43- 

from a fixed point by a string I of which the mass is negligible. 
If the particle P is moved to one side and released it will oscillate 
back and forth. It is desired to show that these oscillations are 
simple harmonic oscillations, and that the period of one complete 
oscillation is equal to 2-k^IJ g, where g is the acceleration of gravity. 
Let Q be the position of the oscillating particle at a given in- 
stant. The length x of the circular arc PQ is equal to lp, and 
the component Qf of the force mg with which gravity pulls 
downwards on the particle, is equal to mg sin or, if the angle 
<p is very small, this force is equal to mg'ip, or to mg/l times cpl, 
or to mg/l times x. Therefore, remembering that the force 
QK = F) is to the left when the arc PQ( = x) is to the right, we 
have 

mg 

7 



F = 



x. 



104 



ELEMENTS OF MECHANICS. 



But this equation is identical in form to equation (15) since mg/l 
is constant, therefore the pendulum bob in Fig. 44 performs sim- 
ple harmonic motion, and the equation expressing the period of 
the oscillations may be found by substituting mg/l for k in 
equation (17). In this way we find 



mg 
I 



4x m 



or 



27T 



i 



(18) 



44. Harmonic motion represented by a curve of sines. If the 

point P', Fig. 45, moves around the circle at uniform speed, the 

angle P'OA is proportional 
to elapsed time, and it may 
be written cot where co is a 
constant and t is elapsed 
time reckoned from the in- 
stant that P' was at A. 
p^— Therefore we may write 




x = r sin co/. 



(19) 



That is, the distance of 
the vibrating particle P 
from its equilibrium posi- 
tion is proportional to 
the sine of a uniformly in- 
creasing angle, and if values of x be plotted as ordinates and 
the corresponding values of t (or oot) as abscissas, we will have a 
curve of sines as shown in Fig. 46. If a fine pointer be attached 
to the prong of a tuning fork, the pointer may be made to trace 
a curve of sines by setting the fork in vibration and drawing 
the pointer uniformly across a piece of smoked glass. 



DYNAMICS. TRANSLATORS MOTION. 



105 




Fig. 46. 



SYSTEMS OF PARTICLES. 

45. System of particles. The ideas of translatory motion may 
conceivably be extended so as to serve as a basis for the descrip- 
tion of any motion of any body or substance, by looking upon the 
body or substance as a collection of particles and considering the 
varying position, velocity, and acceleration of each particle. A 
collection of particles treated in this way is called a system of par- 
ticles or simply a system. Thus a rotating wheel is a system of 
particles, a portion of flowing water is a system of particles, 
a given amount of a gas is a system of particles. The word con- 
figuration is used when we wish to refer to the relative positions 
of the particles of a system; thus the configuration of a system 
is said to change when the particles change their relative positions, 

A closed system is a system no particle of which has any force 
acting on it from outside the system. There is no such thing in 
nature as a closed system, but the conception is useful never- 
theless. 

The cases in which it is not only conceivably possible, but 



106 ELEMENTS OF MECHANICS, 

actually feasible to study more or less complicated types of 
motion by treating the moving substance as a system of particles, 
are as follows: 

(a) The case in which the system consists of very few bodies 
and where each body may be treated as a particle.* This case 
is exemplified by the sun and planets. 

(b) The case in which the particles of a system move in a 
perfectly regular or orderly way. Thus the particles of a ro- 
tating wheel move in an orderly fashion, the particles in a smooth- 
ly flowing liquid move in an orderly fashion, the particles of a 
vibrating string move in an orderly fashion, the connected parts 
of any machine such as a steam engine or a printing press move 
in an orderly fashion. Any system in which orderly motion 
takes place is called a connected system. 

(c) The case in which the particles of a system move in utter 
disorder, without any connection whatever with each other. In 
this case it would evidently be impossible to consider the actual 
motion of each particle, in fact the only possible treatment of 
such a system is a treatment based on the idea of averages and 
probable departures therefrom. Thus the very important kinetic 
theory of gases has been built up on the hypothesis that a gas 
consists of innumerable disconnected particles in disordered 
motion. 

46. Momentum. In the discussion of a system, the product 
mv of the mass of a particle and its velocity is of sufficient im- 
portance to warrant its receiving a name ; it is called the momen- 
tum of the particle, it is a vector and its direction is the same as 
the velocity v of the particle. 

When an unbalanced force acts upon a particle, of course the 
momentum of the particle changes; the rate of change of the 
momentum is equal and parallel to the force. This is evident 
when we consider that a change of velocity Av means a change 
of momentum equal to m-Av, which, divided by the elapsed time 
At, gives the rate of change of momentum; but Av/At is equal 

*Or where each body is a connected system, see (&). 



DYNAMICS. TRANSLATORY MOTION. 107 

to acceleration, so that m . Av/At is equal to mass times acceler- 
ation, and this is equal to the unbalanced force, according to 
equation (1), Art. 33. 

The mutual force-action of two particles cannot change the total 
momentum of the two particles. This is evident when we consider 
that the mutual force-action of two particles consists of two equal 
and ppposite forces (action and reaction) , so that while one par- 
ticle gains momentum in one direction, the other particle gains 
momentum in the opposite direction at the same rate. The con- 
stancy of the total momentum of two particles, insofar as their 
mutual force-action is concerned, is called the principle of the con- 
servation of momentum. 

The principle of the conservation of momentum applies to any number of particles 
insofar as their mutual force-actions are concerned. The total momentum of the 
particles of a system is never changed bj - the mutual force-actions within the system, 
or, in other words, the total momentum of a closed system is constant. 

47. Impact. Consider two particles of which the masses are m± and m 2 , and 
the velocities v v and v 2 , respectively. The combined momentum of the two particles 
is m^-\-m 2 v 2 . If the bodies collide, their velocities may change. Let V x and 
V 2 be the respective velocities after impact. Then m x V l J rm 2 V 2 is the total 
momentum of the bodies after impact, and by the principle of the conservation of 
momentum, we have 

m 1 v 1 -{-m 2 v 2 = miV 1 -\-m 2 V 2 (a vector equation) (i) 

Impact of inelastic balls.— When an inelastic ball, such as a ball of soft clay 
strikes squarely against another, the two balls move after impact as a single body so 
that Vi and V 2 are equal, and this common velocity after impact is completely 
determined by equation (i). 

Impact of perfectly elastic balls. — Consider two elastic balls moving at velocities 
v Y and » a in the same straight line (v { and v 2 being opposite in sign if the balls are 
moving in opposite directions). Let the masses of the balls be m x and m 2 re- 
spectively. 

When these balls collide they are distorted, and at a certain instant the distortion 
reaches a maximum, after which the balls rebound from each other and the distor- 
tion is relieved. When the distortion of the two balls has reached its maximum, the 
two balls are at the instant moving at common velocity c, which is determined by the 
equation 

(mi-\-m 2 )c = m l v 1 -\-m 2 v 2 (ii) 

During the time that the balls are being distorted, which time we shall call the 
first half of the impact, the first ball loses* an amount of velocity (v x — c) and the 
second ball loses* an amount of velocity (v 2 — c). During the time that the balls 

*The velocity c lies between v 1 and v 2 so that if (v x — c) is positive then (v 2 — c) 
must be negative. 



108 ELEMENTS OF MECHANICS. 

are being relieved from distortion, which time we shall call the second half of the 
impact, they are assumed to act on each other with precisely the same series of 
forces as during the first half of the impact, only in a reverse order. This is what is 
meant by the assumption that the two balls are perfectly elastic. Therefore during 
the second half of the impact, each ball loses the same amount of velocity as it lost 
during the first half of the impact, that is, the total loss of velocity by the first ball 
is 2(» x — c) and the total loss of velocity by the second ball is 2{v 2 — c), so that 

V 1 = V 1 — 2(V ] —C) 

and 

V 2 = v 2 — 2O2— c) 
or 

V 1 = 2C — V l (iii) 

and 

V 2 = 2C — V 2 (iv) 

in which V x and V 2 are the respective velocities of the balls after impact. 

Substituting the value of c from equation (ii) in equations (iii) and (iv) we have 

m x v l + 2m 2 v 2 — m 2 v x 

y — - ( v ) 

m x +m 2 

2m 1 v 1 + m 2 v 2 - m x v 2 

V 2 = -7- (vi) 

m 1 + m 2 

The simplest case is where m x — m x and where v 2 = o, that is where the balls are 
similar, and where the first ball only is in motion before impact. In this case the 
result may be derived from equations (v) and (vi) but it is more instructive to de- 
rive the result anew. The common velocity c at the middle of the impact is equal 
to 3^ V\> That is, the first ball loses half its velocity and the second ball gains an equal 
amount of velocity during the first half of the impact. During the second half of 
the impact the first ball loses the remainder of its velocity and comes to a standstill, 
and the second ball gains once more an equal amount of velocity so that its velocity 
is now equal to the initial velocity v x of the first ball. That is, when an elastic ball 
A strikes squarely against a similar stationary ball B, the ball A stops, and the ball 
B moves on with the full original velocity of A. If A is heavier than B, then both 
balls move in the same direction after the impact. If B is heavier than A, then A 
moves backwards, or has a negative velocity after the impact. 

48. Motion of the center of mass of a system. The center of 
mass of a system has been defined in physical terms in Art. 30. 
The center of mass of a body of uniform density is at the geo- 
metrical center of the body. The center of mass of two particles 
lies on the line joining them, and its distance from each particle 
is inversely proportional to the mass of the particle. Thus the 
center of mass of the earth and moon is on the line joining the 
center of the earth and the center of the moon, and it is about 
80 times as far from the center of the moon as it is from the 



DYNAMICS. TRANS LATORY MOTION. IO9 

center of the earth (3,000 miles from the center of the earth), 
inasmuch as the mass of the earth is about 80 times as great as 
the mass of the moon. 

The center of mass of a system remains stationary, or continues 
to move with uniform velocity in a straight line, if the vector sum 
of all of the forces which act on the system is zero. 

For example, consider an emery wheel mounted on a shaft and 
rotating at high speed. If the center of mass of the wheel lies 
in the axis of the shaft, it, of course, remains stationary as the 
wheel rotates, and the only force that need be exerted on the shaft 
by the bearings is the steady upward force required to balance 
the downward pull of the earth on the wheel. A rotating ma- 
chine part is said to be balanced when its center of mass is in its 
axis of rotation. 

When the center of mass of a system is not stationary, and does 
not move with uniform velocity in a straight line, then the vector sum 
F s of the forces which act on the system is not zero. 

In fact, the acceleration A of the center of mass of a system of 
particles, the vector sum F g of the forces which act on the system, 
and the total mass M of the system are related to each other 
precisely in the same way as the acceleration, force, and mass of 
a single particle. That is, as fully explained in Art. 50, we have 

F s = MA (20) 

Example 1. Consider an emery wheel of which the center of 
mass lies at a distance r to one side of the axis of rotation, then, 
as the wheel rotates, the center of mass describes a circular path 
of radius r, the acceleration of the center of mass is equal to 
471-W at each instant, and a side force equal to \ir 2 n 2 rM and 
parallel to r at each instant must act on the axle to constrain the 
center of mass to its circular path, precisely as if the entire mass 
of the wheel were concentrated at its center of mass. 

Example 2. The centrifugal drier consists of a rapidly rotat- 
ing bowl mounted on top of a vertical spindle, and the materials 
to be dried are placed in this bowl. It is impossible to keep the 



IIO ELEMENTS OF MECHANICS. 

bowl and contents even approximately balanced, so that, if the 
spindle were carried in a rigid bearing, the machine would be 
disabled in a short time because of the very great forces that 
would be brought into play in constraining the center of mass of 
bowl and contents to move in a circular path. This difficulty is 
obviated by supporting the spindle at the lower end only, in a 
long bearing mounted on springs to hold it approximately vertical. 
The bowl, contents, and spindle then rotate about a line passing 
through their center of mass and through the center of the flexi- 
ble bearing, and, although the bowl and spindle seem to wobble 
badly (inasmuch as they do not rotate about the axis of figure), 
nevertheless the machine runs quite smoothly, producing but 
little vibration in the supporting frame. 

Example j. If two balls, which are tied together with a short 
string, are thrown in such a way that the string is kept stretched 
while the balls revolve rapidly about one another, a certain point 
of the string will describe a smooth parabolic curve, just as a 
simple projectile would do. This point of the string is the center 
of mass of the two balls. The center of mass of the earth and 
the moon describes an elliptic orbit about the sun once a year, 
while trie earth and moon rotate about their center of mass once 
every lunar month, in a manner very similar to the motion of the 
two balls just described. 

49. The balancing of a rotating machine part. Any part of a 
machine which is to rotate rapidly must be adjusted so that its 
center of mass lies in the axis of rotation.* This adjustment is 
called balancing, and a machine part so adjusted is said to be 
balanced. A machine part which is to be balanced, a dynamo 
armature for example, is mounted on its shaft and the ends 
of the shaft are placed upon two straight level rails. If the 
center of mass is in the axis of the shaft, the whole will stand in 
equilibrium in any position; whereas, if the center of mass is not 

*A machine part which is long in the direction of the shaft upon which it rotates 
may have its center of mass in the axis of the shaft and yet the bearings may have 
to exert constraining forces upon the shaft as the part rotates. A long cylinder 
loaded on opposite sides at the two ends is an example. 



DYNAMICS. TRANSLATORY MOTION. 



Ill 



in the axis of the shaft, the whole will come to rest with the 
center of mass at the lowest possible position, and material 
is removed from one side or added to the other side until the 
center of mass is in the center of the shaft. 

Figure 47 shows a wheel mounted on a pair of balancing rails. 
Such a pair of balancing rails is a prominent feature in a shop 
where the fly-wheels of large engines have to be balanced. 




Fig. 47- 



50. Equations of center of mass. The position of the center of mass of a system 
of particles may be expressed in terms of the positions and masses of all of the par- 
ticles in the system as follows : Let x be the ^-coordinate of a particle whose mass is 
m, let x' be the ^-coordinate of a particle whose mass is m' ', let x" be the x-coordinate 
of a particle whose mass is m" and so on, then the sum mx-\-m , x , -\-m"x"-\-z\.c> 
divided by the total mass of the system, namely, ra+w'+ra"-f-etc, gives the 
x-coordinate of the center of mass of the system. That is, the x-coordinate of 
the center of mass is 

2mx 
2m 



X = 



(21) 



and exactly similar expressions may be formulated for the y-coordinate and for the 
z-coordinate of the center of mass. 

If the origin of coordinates is at the center of mass of the system then, of course, 
X is equal to zero, and equation (21) becomes 

Srax = (22) 



112 ELEMENTS OF MECHANICS. 

In order to show that equation (20) is true, it is sufficient to consider only the x- 
component of A, and the ^-components of the accelerations of the respective 
particles. The sc-component of A is d l X\dt l and the ^-components of the accel- 
erations of the respective particles are d 2 x\dt 2 , d 2 x'ldt 2 , d 2 x"ldt 2 and so on. There- 
fore, writing M for 2m in equation (21), and differentiating twice with respect to 

time, we have 

d?X d 2 x d 2 r/ d 2 x" 

+ etc., 



.d 2 X 


d*x . ,d 2 x f , dV' 


dt l 


==m d7 +m ~d¥ + -^ 



but m{d 2 x\dt 2 ) is the x-component of the force acting on the particle ra, m , (d 2 x , ldt 2 ) 
is the x-component of the force acting on the particle m' and so on, so that the right" 
hand member of this equation is the sum of the x-components of all the forces acting 
on the particles of the system, and this is equal to the sum of x-components of all 
of the external forces acting on the particles of the system, inasmuch as mutual force- 
actions between the particles of the system cancel out of this sum because such 
mutual force-actions consist of pairs of equal and opposite forces. Therefore, the 
right hand member of the above equation is the x-component of the total external 
force F s which acts on the system and the above equation reduces to 

M times x-component of A = x-component of F s (i) 

and we may show in exactly the same way that 

M times y-component of A = y-component of F s (ii) 

and 

M times z-component of A = z-component of F s (iii) 

These three equations are equivalent exactly to the single vector equation (20). 

Problems. 

52. A train having a mass of 350 tons (2,000 pounds) starting 
from rest reaches a speed of 50 miles per hour in 2^ minutes. 
What is the average pull of the locomotive during 2}4 minutes, 
dragging forces of friction being neglected? Ans. 10,700 pounds- 
weight. 

53. The above train moving at a speed of 50 miles per hour is 
brought to a standstill in 16 seconds by the brakes. What is the 
average retarding force in pounds-weight due to the brakes? 
Ans. 100,300 pounds- weight. 

54. An elevator reaches full speed of 8 feet per second 2}4 
seconds after starting. With what average force in pounds- 
weight does a 160-pound man push down on the floor while the 
elevator is starting up? The elevator is stopped (when moving 
up at full speed) in 1 }4 seconds. With what average force in 
pounds-weight does a 160-pound man push down on the floor 



DYNAMICS. TRANSLATORS MOTION. 113 

while the elevator is stopping? Ans. 176 pounds-weight while 
the elevator is starting up; 133.3 pounds- weight while the ele- 
vator is stopping. 

Note. In the first case the upward push of the floor on the man exceeds the 
weight of the man by the amount which is necessary to produce the upward accelera- 
tion; in the second case the weight of the man exceeds the upward push of the floor 
by the amount which is necessary to produce the downward acceleration. 

The use of D'Alembert's principle simplifies the argument of this problem greatly 
Consider the case in which the elevator is gaining velocity upwards. In this case 
an unbalanced upward force equal to 1/32.2 • ma must be acting on the man. There- 
fore introducing a fictitious downward force equal to 1/32.2 • ma, and proceeding as 
in a problem in statics, we consider that the upward force exerted on the man by the 
platform must be equal to the total downward force acting on the man, namely, 
the weight of the man plus the fictitious downward force of 1/32. 2ma. D'Alembert's 
principle is stated on page 64. 

55. An elevator car has a mass of 1,000 pounds. It gains a 
velocity upwards of 8 feet per second in 2^ seconds after start- 
ing from rest. Calculate (a) the tension on the rope while the 
car is stationary, (b) the average tension of the rope while the 
car is starting upward, and (c) the tension of the rope while the 
car is moving at the full speed of 8 feet per second. Ans. (a) 
1,000 pounds-weight; (b) 1,100 pounds- weight; (c) 1 ,000 pounds- 
weight. 

56a. Find the tension of the rope required in problem 46 to 
produce an upward acceleration of 2 feet per second per second 
of the elevator car. Ans. 1596.9 pounds-weight. 

Note. The unbalanced upward force necessary to produce the specified accel- 
eration, as calculated by equation (5), is 92.5 pounds-of-force, and the point of 
application of this upward unbalanced force is the center of mass C in Fig. 46^. 
The simplest method of solving this problem is to reduce it to a problem in statics 
by means of D'Alembert's principle. A downward force of 92.5 pounds may be 
thought of as acting at the point C in addition to the weight of the car, and then the 
first and second conditions of equilibrium may be applied exactly as in problem 46. 

56b. Find the downward acceleration of the car in problem 
56a when the tension of the rope is 1450 pounds-weight. Ans. 
1.032 feet per second per second. 

57. A train having a mass of 1,200 tons (2,000 pounds) is to 
be accelerated at }4 mile per hour per second up a j4 per cent, 
grade. The train friction is 10 pounds per ton. Find the neces- 
9 



114 ELEMENTS OF MECHANICS. 

sary draw-bar pull of the locomotive. Ans. 79,000 pounds- 
weight. 

Note. A }/2 per cent, grade is one that rises 3^ foot in 100 feet of horizontal 
distance. 

58. A cord is strung over a pulley. At one end of the cord 
is a 10 pound body, and at the other end of the cord is a 11 
pound body. Neglecting the weight of the cord and the fric- 
tion and mass of the pulley, find the acceleration of each body 
and the tension of the cord. Ans. 1.523 feet per second per 
second; tension of cord 10.476 pounds-weight. 

59. A falling ball passes a given point at a velocity of 12 feet 
per second. How far below the point is the ball after 5 seconds? 
How far does the ball fall during the fifth second after passing 
the given point? Air friction neglected. Ans. After 5 seconds 
the ball is 460 feet below the point, during the fifth second the 
ball falls 156 feet. 

60. A heavy iron ball is tossed at a velocity of 20 feeet per 
second in a direction 30 above the horizontal. What are its 
horizontal and vertical distances from the starting point after ^ 
second? Air friction neglected. Ans. Horizontal distance 12.99 
feet; vertical distance — 1.5 feet. 

Note. Find vertical and horizontal components of the initial velocity. The 
latter component remains unchanged while the vertical motion of the ball is pre- 
cisely what it would be if it had no horizontal motion. 

61. A heavy shot is thrown in a direction 30 above the hori- 
zontal, it strikes the ground 50 feet from the thrower, and the 
shot is 5 yi feet above the ground when it leaves the thrower's 
hand. What is the initial velocity v of the shot? Air friction 
neglected. Ans. 39.4 feet per second. 

Note. The horizontal velocity, v cos 30 , is constant, the time of flight in seconds 
is * = 5o feetn-(z) cos 30 ), and the vertical distance fallen, namely sVl feet, is equai 
to v sin 3oXt+}4gt 2 , in which t is the time of flight in seconds. The acceleration 
of gravity is here to be considered as negative. 

62. An 80-ton (2,000 pounds) locomotive goes round a rail- 
way curve of which the radius is 600 feet at a velocity of 65 feet 
per second. With what force in pounds-weight do the flanges 



DYNAMICS. TRANSLATORS MOTION. 115 

of the wheels of the locomotive push against the outer rail when 
the outer rail is not elevated? Ans. The flanges of the locomotive 
exert a horizontal force on the outer rail of 35,210 pounds- weight. 

63. Calculate the proper elevation to be given to the outer rail 
on a railway curve of 600 feet radius for a train speed of 65 feet 
per second, the width of the track being 4 feet S}4 inches. Ans. 
1. 01 5 feet. 

64. The tension of a belt is 50 pounds-weight. With what 
force in pounds-weight does the belt push against each inch of 
circumference of a pulley 12 inches in diameter when the pulley 
is stationary? Ans. 8.33 pounds- weight per inch of circum- 
ference. 

Note. The static relation between tension in a circular hoop and actual out- 
ward forces acting on each part of the hoop is the same as the relation between ten- 
sion and the unbalanced inward forces in the case of a rotating hoop as discussed 
in Art. 40. 

65. The mass of each inch of length of the belt specified in 
problem 64 is 0.24 pound. With what force in pounds- weight 
does the belt push inwards against each inch of circumference of 
the 12-inch pulley when the pulley revolves at a speed of 300 
revolutions per minute, the tension of the belt being 50 pounds- 
weight? Ans. 4.47 pounds- weight per inch of circumference. 

Note. The tension of the belt is capable of producing an inward force of 8.33 
pounds-weight per inch of circumference of the wheel when the pulley is stationary. 
When the pulley is rotating the inward force exerted on each inch of circumference 
of the pulley is reduced by an amount equal to the unbalanced force which must 
pull inwards on each inch of length of the belt according to equation (10) in Art. 38: 

66. The car next to the locomotive in a train is 35 feet long 
between bumpers and it is pulled at each end with a force of 
10,000 pounds (the force at the rear end of the car is of course 
somewhat less than the force at the front end). The train rounds 
a circular curve of 1,000 feet radius at a speed of 20 miles per 
hour. The car with its load weighs 100,000 pounds. Find the 
horizontal force, at right angles to the track, with which the track 
acts on the car. Ans. 2,339 pounds- weight. 

Note. The portion of a train directly behind the locomotive is under tension 
like a belt, and the tension helps to constrain the cars to their circular path exactly 



Il6 ELEMENTS OF MECHANICS. 

as in the case of a belt passing around a pulley. In solving this problem it is suffi- 
ciently accurate to use the formula F= Tjr in which T is the tension of a belt, r is 
the radius of the circular arc formed by the belt, and F is the radial force per unit 
length of belt due to T. 

67. A force of 5 X io 6 dynes deflects the end of the spring in 
Fig. 43 through a distance of 1.25 centimeters. What is the 
value of the constant k in equation (15), and in terms of what 
unit is this constant expressed? How much force would be 
required to deflect the end of the spring through a distance of 2 
centimeters? Ans. The value of k is 4 million dynes per centi- 
meter; 8 million dynes. 

68. A mass of 2 kilograms is attached to the end of the spring 
specified in problem 6j, and the mass is set vibrating. How 
many complete vibrations will it make per minute? Ans. 427 
vibrations per minute. 

69. A force of 10 pounds-weight deflects the end of the spring 
in Fig. 43 through a distance of 0.02 foot. What is the value of 
the constant k in equation (15) and in terms of what unit is this 
constant expressed? A mass of 10 pounds is attached to the 
end of the spring, how many complete vibrations will the 10 
pound mass make per minute? Ans. The value of k is 16,000 
poundals per foot; 382 vibrations per minute. 

70. What is the length I of a simple pendulum which makes 
one complete vibration per second at a place where the accelera- 
tion of gravity is 981 centimeters per second per second? Ans. 
24.85 centimeters. 

71. A wheel has a mass of 50 pounds, its center of mass is 
0.2 inch from the axis of the shaft upon which the wheel rotates, 
and the speed of the wheel is 600 revolutions per minute. How 
much force in pounds-weight must act on the shaft to constrain 
the center of mass to its circular path? What is the direction of 
the force at each instant? Ans. 102.8 pounds- weight from the 
center of mass towards the axis of the shaft. 

72. A ballistic pendulum AB, Fig. 72^?, is suspended by two 
cords ss, the length of each of which is 400 centimeters, and the 
body A B has a mass of 10 pounds. A rifle bullet of which the mass 



$y///m, 


V%).'////M//Mb 




8 


— r -- 
i 


8 


i R" 


1 L, 


' __ j 


>'y=- 




l TT,l..l1 



DYNAMICS. TRANSLATORY MOTION. 117 

is 0.005 pound, strikes AB at the point indicated by the short 
arrow, and the velocity imparted to AB carries it through a 
horizontal distance of 8 inches before 
it is brought to rest by gravity. Find 
the velocity of the bullet. The accel- 
eration of gravity is 32 feet per sec- 
ond per second. Ans. 2,264 ^ eet P er 
second. 

Note. The center mass of A B describes the 
arc of a circle of which the radius is I. Calculate Fig- 7 2 P- 

the vertical movement of AS from the known 

value of I and the specified horizontal movement of AB . Then calculate the ve- 
locity of A B which would suffice to lift A B through this vertical distance, and 
then calculate the velocity of the bullet by using the principle of the conservation 
of momentum. 

73. A ball weighing 550 pounds is shot from a 150,000 pound 
gun at a velocity of 2,500 feet per second. What is the back- 
ward velocity of the gun as the ball leaves the muzzle? Sup- 
pose the gun is allowed to move back two feet during the recoil, 
what is the average value of the force required to bring it to rest? 
Ans. 9.17 feet per second; 98,540 pounds-weight. 

74. An ivory ball of which the mass is 500 grams, and of which the velocity is 
100 centimeters per second, collides with a stationary ivory ball of which the mass is 
1,000 grams, the line connecting the centers of the balls being parallel to the velocity 
of the moving ball. Find the common velocity of both balls after their relative 
motion has been reduced to zero during the first half of the impact, and find the 
velocity of each ball after impact; specify direction of each velocity. Ans. Common 
velocity 33.3 feet per second; —33.3 feet per second is the velocity of the small ball 
after impact, and +66.7 feet per second is the velocity of the large ball after impact. 

Note. Assume that the ivory balls are perfectly elastic as explained in Art. 47. 



CHAPTER V. 

FRICTION. WORK AND ENERGY. 

51. Friction. A body in motion is always acted upon by 
dragging forces which oppose its motion and tend to bring it to 
rest. This action is called friction. 

Sliding friction. When one body slides on another the motion 
is opposed by a frictional drag. Thus the cross-head of a steam 
engine slides back and forth on the guides, a rotating shaft slides 
in its bearings, and the motion is in each case opposed by a fric- 
tional drag. 

Fluid friction. The flow of water through a pipe or channel, 
the motion of a boat, and the motion of a projectile through the 
air are opposed by friction. This type of friction is called fluid 
friction and it is discussed in a subsequent chapter. 

Rolling friction. The frictional drag upon a wheeled vehicle is 
due in part to the sliding friction at the journals, in part to the 
friction of the air, and in part to the continual yielding of the road 
or track under the wheels. The effect of this yielding is very 
much as if the vehicle were continually going up a hill, the top 
of which is never reached. The frictional drag on a wheeled 
vehicle due to the yielding of the road or track is sometimes 
called rolling friction. 

Frictional drag due to unevenness of a road bed. When a 
vehicle is (Jrawn very slowly over a rough road, the wheels roll 
"up hill," as it were, when they strike a small stone and then 
"down hill" again when they leave the stone, and the average 
value of the pull required to draw the vehicle is not effected by 
unevenness of road bed; but if the speed of the vehicle is 
great, the unevenness of the road bed produces a very consid- 
erable frictional drag, the effect is as if the wheels were being all 
the time "rolled up" a succession of small hills not to "roll 

118 



FRICTION. WORK AND ENERGY. 



II 9 



down" again, but to come down each time with a bump. This 
kind of friction shows itself in the vibration and swaying of a 
vehicle, and it is one of the most prominent causes of frictional 
drag upon a vehicle which is driven at high speed. 

52. Coefficient of sliding friction. The horizontal force H 
required to cause a body to slide steadily over the smooth hori- 
zontal surface of another body is approximately proportional to 
the vertical force V which pushes the body against the surface. 

That is 

H = nV (23) 

in which V is the force with which a body is pushed against any 
smooth surface, and H is the force, parallel to the surface, which 
causes the body to slide. The proportionality factor fi is called 
the coefficient of friction; it is nearly independent of the contact 
area of the sliding substances and it does not vary greatly with 
the velocity of sliding. Thus the coefficient of friction of wood 
on a smooth metal surface is about 0.40, the coefficient of fric- 
tion of smooth brass on smooth steel (not oiled) is about 0.22. 

Angle of sliding friction. Consider a block B, Fig. 48a, sliding 

H ^ ^ H 





on a table TT in the direction of the dotted arrow; let V be the 
force with which the block is pushed against the table and let H 
be the force necessary to keep the block in motion. Then since 
the ratio H/V is constant (that is, if V is large, H is large in 
proportion) , it is evident that the angle <p between V and the resul- 
tant force F is constant. This angle is called the angle of friction 
of the given substances B and T, and evidently the tangent of 
<p is equal to the coefficient of friction /x of the sliding substances. 




120 ELEMENTS OF MECHANICS. 

Figure 486 represents the table TT tipped so as to bring the 
force F into a vertical position. In this case the force F may be 
thought of as the pull of gravity on the block B, and the com- 
ponent of F parallel to the table (namely H) is barely sufficient 
to cause the block to slide. 

It is important to notice that the force F in Fig. 48a is the total 
force which the sliding block exerts upon the table. Consider 

blocks A A and BB in contact as 
shown in Fig. 48c. The block A can 
exert upon the block B a force in any di- 
rection provided the line of action of the 
force lies inside of a cone of which the 
half -angle is <p and of which the axis is 
normal to the surface of contact of A A 
and BB, where <p is the angle of 
Fi 8c friction of the two substances A and 

B. 
The friction between the two sliding surfaces is approximately 
in accordance with the above statements when the surfaces are 
smooth and made of unlike materials. Thus wood sliding on 
metal, polished steel sliding on brass or Babbitt metal, and hard 
steel sliding on the polished surface of a jewel, all have fairly 
well defined coefficients of friction. When the sliding surfaces 
are rough, however, there is no regularity whatever in the friction, 
and when the substances are similar the friction is sometimes 
very irregular even though the surfaces are smooth. Thus brass 
on brass tends to weld and tear in a most remarkable manner 
and a clean plate of glass cannot be made to slide over another 
clean plate of glass at all (if the surfaces are very clean) unless 
there is an air cushion between them. 

53. Active forces and inactive forces. Definition of work. 

Nothing is more completely established by experience than the 
necessity of employing an active agent such as a horse or a steam 
engine to drive the machinery of a mill or factory, to draw a car, 
or to propel a boat ; and although the immediate purpose of the 



FRICTION. WORK AND ENERGY. 121 

driving force may be described in each case by saying that the 
driving force overcomes or balances the opposing forces of fric- 
tion, still the fact remains that the operation of driving a machine 
or propelling a boat involves a continued effort or cost. Indeed 
to supply a man with the thing (energy) which will drive his mill 
or factory, is to supply him with a commodity as real as the 
wheat he grinds or the iron which he fabricates into articles of 
commerce. Wheat and iron are sharply defined as commodities 
in the popular mind on the basis of many generations of com- 
mercial activity, because wheat and iron can be stored up and 
taken from place to place, and because change of ownership is so 
easily accomplished and so simply accounted for. That which 
serves to drive a mill or factory, however, cannot be stored up 
except to a very limited extent, and it is only in recent years that 
means have been devised for transmitting it from place to place 
and that an exact system of accounting has been established for 
governing its exchange. A clear idea of energy does not exist 
as yet in the popular mind, and the following definitions cannot 
be expected to convey a full and clear idea at once. 

The common feature of every case in which motion is main- 
tained is that a force is exerted upon a moving body and in the 
direction in which the body moves. Such a force is called an 
active force,* and to keep up an active force requires continuous 
effort or cost. 

A force which acts on a stationary body, on the other hand, 
may be kept up indefinitely, without cost or effort; and such a 
force is called an inactive force. Thus a weight resting on a 
table continues to push downward on the table, a weight sus- 
pended by a string continues to pull on the string, the main 

*An active force is any mutual force action between two bodies one of which 
moves with respect to the other. The force with which a boy pulls on a sled is 
called an active force although there is no relative motion between the boy and the 
sled. In this case, however, the work is actually done in the boy's legs and the 
active force in the boy's legs is exerted against the ground which moves backwards 
with respect to the boy. To push on the front door of a moving car is not to exert 
an active force. 



122 ELEMENTS OF MECHANICS. 

spring of a watch will continue indefinitely to exert a force upon 
the wheels of the watch if the watch is stopped. 

The idea of an inactive force is applicable also to a force which 
acts on a moving body but at right angles to the direction in 
which the body moves. Thus the vertical push of a driver on 
the seat of a wagon which travels along a level road is an inactive 
force, the forces with which the spokes of a rotating wheel pull 
inwards on the rim of the wheel are inactive forces. 

An active force is said to do work, and the amount of work 
done in any given time is equal to the product of the force and 
the distance that the body has moved in the direction of the force. 
That is 

W = Fd (24) 

in which F is the force acting on a body, and W is the work 
done by the force during the time that the body moves a dis- 
tance d in the direction of F. If d is not parallel to F, then 
W = Fd cos 6, where is the angle between F and d. 

Units of work. The unit of work is the work done by unit 
force while the body, upon which the force acts, moves through 
unit distance parallel to the force. 

The erg, which is the c. g. s. unit of work, is the work done 
by a force of one dyne while the body, upon which the force acts, 
moves through a distance of one centimeter in the direction of 
the force. The erg is, for most purposes, inconveniently small, 
and a multiple of this unit, the joule, is much used in practice. 
The joule* is equal to ten million ergs (io 7 ergs). 

The work done by a force of one pound-weight while the 
body upon which the force acts moves through a distance of one 
foot in the direction of the force, is called the foot-pound, f 

*It is frequently convenient to have a name for that unit of force which multiplied 
by one centimeter gives one joule of work, according to equation (24). This unit 
of force may be called the joule per centimeter. 

tThe kilogram-meter is the work done by a force of one kilogram-weight while 
the body upon which the force acts moves through a distance of one meter in the 
direction of the force. The foot-pound unit of work is used quite generally by 
American and English engineers, and the kilogram-meter unit of work is used in 
those countries where the metric system has been adopted. 



FRICTION. WORK AND ENERGY. 1 23 

54. Power. The rate at which an agent does work is called 
the power of that agent. Thus a locomotive exerts a pull of 
15,000 pounds- weight on a train and draws the train through a 
distance of 500 feet in 10 seconds. The work done is 7,500,000 
foot-pounds which, divided by the time interval of ten seconds, 
gives 750,000 foot-pounds per second, as the rate at which the 
locomotive does work. 

Units of power. Power may, of course, be expressed in ergs 
per second, in joules per second, or in foot-pounds per second. 
The unit of power, one joule per second is called a watt. The 
horse-power, which is extensively used by engineers, is equal to 
746 watts or to 550 foot-pounds per second. 

Power developed by an active force. Consider a force F acting 
upon a body which moves in the direction of the force at velocity 
v. During t seconds the body moves through the distance vt and 
the amount of work done is F X vt according to equation (24), 
and, dividing this amount of work by the time, we have 

P = Fv (25) 

in which P is the power developed by an active force F, and v is 
the velocity with which the body, upon which F acts, moves in 
the direction of F. If F is expressed in dynes and v in centime- 
ters per second, then P is expressed in ergs per second; if F is 
expressed in pounds- weight and v in feet per second, then P is 
expressed in foot-pounds per second. 

Example. A horse pulls with a force of 200 pounds weight 
in drawing a loaded cart at a velocity of 3 feet per second and 
develops 600 foot-pounds per second of power. 

Measurement of power. Nearly all practical measurements 
relating to work are measurements of power. The power of an 
agent may be measured as follows: 

(a) The value of an active force and the velocity of the body 
upon which the force acts may be measured and the power may 
then be calculated according to equation (25). 

Examples. (1) The draw-bar pull of a passenger locomotive 



124 ELEMENTS OF MECHANICS. 

is measured by means of a heavy spring scale and found to be 
6,000 pounds, and the velocity of the locomotive, as determined 
by the distance traveled in a given time, is found to be 90 feet per 
second. From these data the net power developed by the loco- 
motive (not counting the power required to propel the locomo- 
tive itself) is found to be 540,000 foot-pounds per second, or 
991 horse- power. 

(2) Let a be the area in square inches of the piston of a steam 
engine, let p be the average steam pressure in the cylinder in 
pounds per square inch as measured by a steam-engine indicator, 
let I be the length of stroke of the piston in feet, and let n be the 
number of revolutions per second made by the engine. Then 
the average force pushing on the piston is pa pounds-weight, 
and the work done during a single stroke is pa Xl foot-pounds, 
and since the number of single strokes per second is m, the 
power developed by the steam is pal X 2w, or 2paln foot-pounds 
per second. The power of an engine determined in this way is 
called its indicated power, to distinguish it from the power de- 
livered by the engine to the machinery which it drives. The 
power delivered by an engine is always less than its indicated 
power on account of frictional losses in the engine. 

(3) An engine to be tested is loaded by applying a brake to its 
flywheel; the pull on the brake (reduced to the circumference 
of the fly-wheel) is 200 pounds- weight ; the velocity of the cir- 
cumference of the flywheel, as determined from the measured 
diameter of the wheel and its observed speed in revolutions per 
second, is 80 feet per second; and the power developed by the 
engine is equal to 200 pounds X 80 feet per second, which is 
equal to 16,000 foot-pounds per second or 29 horse-power. The 
power of an engine determined in this way is called its brake power. 

Figure 49 shows the arrangement of a brake for measuring the 
power of an engine, or of any agent, like an electric motor or 
water wheel, which delivers power from a pulley. The spring 
scale 6 1 measures the force at the end of the brake arm, and this 
observed force is multiplied by a/r to find the equivalent force 



FRICTION. WORK AND ENERGY. 



125 



at the surface of the pulley, where a is the length of the arm as 
shown in Fig. 49 and r is the radius of the pulley. 

(b) Power is frequently measured electrically. Thus the power 
in watts delivered by a direct-current dynamo is equal to the 
product of the electromotive force of the dynamo in volts and the 
current in amperes delivered by the dynamo. 

Power-time units of work. Inasmuch as nearly all practical 
measurements relating to work are measurements of power, it 




Fig. 49. 



has come about that a given amount of work is often expressed 
as the product of power and time. The watt-hour is the amount 
of work done in one hour by an agent which does work at the 
rate of one watt, the kilowatt-hour is the amount of work done in 
one hour by an agent which does work at the rate of one kilowatt 
(one kilowatt is 1,000 watts), and the horse-power-hour is the 
amount of work done in one hour by an agent which does work 
at the rate of one horse-power. 

Efficiency. The efficiency of a machine, like a water wheel, 
a steam engine, a dynamo, or a motor, which transforms energy, 
is defined as the ratio of the power developed by the machine to 
the total power delivered to the machine. 



126 ELEMENTS OF MECHANICS. 

ENERGY. 

55. Definition of energy. Limits of the present discussion. 

Any agent which is able to do work is said to possess energy, and 
the amount of energy an agent possesses is equal to the total 
work the agent can do. Thus the spring of a clock when it is 
wound up is in a condition to do a definite amount of work and 
it is therefore said to possess a definite amount of energy. 

In developing the idea of energy it is important to distinguish 
between an agent which merely transforms energy and an agent 
which actually has within itself the ability to do a certain amount 
of work. Thus the steam engine merely transforms the energy 
of fuel into mechanical work, and a water wheel merely trans- 
forms the energy of an elevated store of water into mechanical 
work, whereas a clock spring, when wound up, has a store of 
energy within itself. 

Whenever a substance or a system of substances gives up 
energy which it has in store, the substance or system of substances 
always undergoes change. Thus the fuel which supplies the 
energy to a steam engine and the food which supplies the energy 
to a horse, undergo chemical change; the steam which carries the 
energy of the fuel from the boiler to the engine cools off or under- 
goes a thermal change when it gives up its energy to the engine; 
a clock spring changes its shape as it gives off energy; an ele- 
vated store of water changes its position as it gives off energy; 
the heavy fly wheel of a steam engine does the work of the 
engine for a few moments after the steam is shut off and the fly 
wheel changes its velocity as it gives off its energy. 

Not only does a substance undergo a change when it gives up 
energy by doing work, but a substance which receives energy or 
has work done upon it undergoes a change. Thus when air is 
compressed by a bicycle pump, work is done on the air and it 
becomes warm; the work done in keeping up the motion of any 
machine or device produces heat at the places where friction 
occurs; when a clock spring is wound up it stores energy by 
its change of shape; when water is pumped into an elevated tank 



FRICTION. WORK AND ENERGY. 1 27 

it stores energy by its change of position; a large part of the 
.work which is expended on a heavy railway train at starting is 
stored in the train by its change of velocity. 

We are now facing a very important (question ; shall we attempt 
a complete discussion of the whole theory of energy at once by 
examining into all kinds of changes which take place when a 
substance does work or has work done upon it; or shall we 
base our discussion on one thing at a time? Most assuredly the 
latter. Therefore let us proceed to discuss the energy relations 
involved in purely mechanical changes, namely, changes of- 
position, changes of velocity, and changes of shape,* and let us 
exclude everything else from our present discussion such as 
chemical changes and thermal changes. 

In attempting to exclude thermal changes from our present 
discussion, however, we are confronted by the fact that friction 
(with its accompanying thermal changes) is always in evidence 
everywhere; and it requires a very high degree of analytical 
power to think only of purely mechanical changes in the face of 
such a fact. This necessary feat of mental effort is greatly facili- 
tated by the use of the idea of a frictionless system; and this 
term will be used whenever it is desired to direct the reader's 
attention exclusively to the energy relations that are involved in 
purely mechanical changes. 

Before proceeding to a minute examination into the mechanical 
theory of energy, it is desirable to establish the ideas of kinetic 
energy and potential energy on the basis of general experience. 
Suppose that a post, standing beside a railway track, is to be 
pulled out of the ground ; can a car-load of stone be made to do 
the work? Certainly it can. All that is necessary is to have 
the car moving past the post and to throw over the post a loop 
of cable which is attached to the moving car. A moving car is 
able to do work; and when it does work its velocity is reduced, 
and its store of energy decreased. The energy which a body 

♦Changes of shape are discussed in Chapter VII. 



128 ELEMENTS OF MECHANICS. 

stores by virtue of its velocity is called the kinetic energy of the 
body. 

It is also a familiar fact that a weight can drive a clock, but in 
doing so the position of the weight changes and its store of en- 
ergy is reduced. The energy which a body stores by virtue of 
its position is called the potential energy of the body. 

The physical reality which lies behind the terms kinetic energy 
and potential energy can perhaps be shown most clearly by con- 
sidering a bicycle rider. Suppose that the rider faces a steep hill 
or a sandy stretch of road where he is called upon to do an un- 
usual amount of work. Every bicycle rider realizes the ad- 
vantage of having a large velocity in such an emergency. This 
advantage of velocity is called kinetic energy.* Or suppose that 
a bicycle rider wishes to use his whole strength, or more if he 
had it, in covering a certain distance; every bicycle rider realizes 
the advantage of being on top of a hill in such an emergency; 
this advantage of position is called potential energy. 

56. Kinetic energy of a particle. The kinetic energy of a 
particle is given by the equation 

W = y 2 mv 2 (26) 

in which W is the kinetic energy in ergs (or foot-poundals) , m 
is the mass of the particle in grams (or pounds), and v is its 
velocity in centimeters per second (or in feet per second). 

Proof of equation (26). The kinetic energy of a particle may 
not only be defined as the work it can do when stopped, but 
also as the work required to establish its motion. Let a con- 
stant unbalanced force F act upon a particle of mass m, then 

F = ma. (i) 

After t seconds the velocity gained is 

v = at (ii) 

*Of course a body can have velocity only in relation to another body and the idea 
of kinetic energy is an idea which applies strictly to a system of particles but not to 
an individual particle. The velocity in equation (26) is velocity referred to the 
earth. 



FRICTION. WORK AND ENERGY. 1 29 

and the distance traveled is 

d = y 2 at 2 . (iii) 

as explained in Art. 35. Therefore, multiplying equations (i) 
and (iii), member by member, we have 

Fd = y 2 maH 2 , (iv) 

but Fd is equal to the work done on the particle and a 2 t 2 is equal 
to v 2 , according to equation (ii), so that equation (iv) reduces to 
W — %mv 2 . 

The kinetic energy of a system of particles is, of course, equal 
to the sum of the kinetic energies of the individual particles of 
the system. 

When mass is expressed in pounds and velocity in feet per 
second, then the kinetic energy of a particle in foot-pounds is 
given, approximately, by the equation 

W =- r — mv 2 . (27) 

64.4 

57. Potential energy. The 'energy stored in a system by virtue 
of the configuration of the system, that is, by virtue of the rela- 
tive positions of the parts of the system, is called the potential 
energy of the system. For example a weight stores energy by 
virtue of its position relative to the earth; a bent spring stores 
energy by virtue of its elastic distortion. 

It is impossible to assign a definite amount of potential energy 
to a system which has a given configuration, for it is impractic- 
able to assign a definite limiting configuration beyond which the 
system cannot go. Thus the weight of a clock might have its 
available store of potential energy increased by boring a hole in 
the clock case so that the weight would move down to the floor, 
then a hole could be bored in the floor and eventually a deep 
well could be dug in the ground. In order to be able to speak 
definitely of the potential energy of a weight it is necessary, there- 
fore, to assign an arbitrary zero position and to reckon the poten- 
10 



130 ELEMENTS OF MECHANICS. 

tial energy of any other given position as the work the weight can 
do in changing from the given position to the chosen zero position. 
In general the potential energy of any system in a given con- 
figuration may be defined as the amount of work the system can 
do in changing from the given configuration to an arbitrarily 
chosen zero configuration. 

Conservative systems. A system (frictionless) which does the 
same amount of work in passing from one configuration to an- 
other, whatever the intermediate stages may be through which 
the system passes, is called a conservative system, and the idea of 
given position^ potential energy applies only to 
such systems. Suppose, for ex- 
ample, that a weight would do 
more work in moving from a given 
position to its chosen zero position 
over one path A than over another 
path B, see Fig. 50a; then the po- 
~ pa tential energy of the weight in the 

Fig- 5oa. . . . f 

given position would be indefinite ; 
and if the weight were carried around the closed path A B in the 
direction of the arrows, then a large amount of work would be 
done in passing down path A and only a portion of this work 
would be required to carry the weight back to the given position 
over path B. That is, work would be created every time the 
weight completed the cycle of motion around A B, and we would 
have "perpetual motion," that is a machine which would do work 
without suffering any permanent change of any kind.* All physi- 
cal systems are conservative in so far as purely mechanical changes 
are concerned; and experience shows that all physical systems 
are conservative when changes of all kinds, mechanical, chemical, 
thermal and electrical, are taken into account; that is, the 

*The idea involved in this discussion of Fig. 50a may be strengthened by introduc- 
ing the idea of cheating to which it stands in clear apposition. Suppose one were to 
hold a weight in his hand and allow it to move downwards in full view of a class, 
and then bring it again to its former position by passing it behind his back where it 
is out of sight with the idea of avoiding the doing of work! 




FRICTION. WORK AND ENERGY. 



131 



energy that a system gives off when it undergoes any change 
whatever, depends only upon the initial and final states of the 
system, and is independent of the intermediate stages through 
which the system may be made to pass. 

Perpetual motion impossible. A perpetual motion machine 
would be a device which would furnish a continuous supply of 
energy for driving machinery. Most of the attempts to produce 
perpetual motion have been quite ridiculous, but on the other 




Fig. 50&. 



Fig. 50c. 



hand many attempts have been quite reasonable. The reason- 
able attempts have nearly all been attempts to get more work out 
of a weight while it falls along one path than is required to carry 
the weight back to its starting point along another path. Figures 
50& and 50c show two perpetual motion devices which were pro- 
posed and tried about 1750. Figure 50& is a rachet wheel to 
which a number of hinged arms are attached, each arm carrying 
a heavy weight. Fig. 50c is a wheel to the rim of which a 
number of bent tubes are attached, each tube containing mercury. 
The arrows show the directions in which the wheels were expected 
to be driven by the increased leverage of the falling weights or 
of the falling mercury. 



132 ELEMENTS OF MECHANICS. 

58. Mutual relation between the kinetic energy and the poten- 
tial energy of a closed system. The most clearly intelligible 
statement of the idea as to the impossiblity of perpetual motion 
may be made by considering an ideal system upon which no 
forces act from the outside. During the motion of such a system 
no work would be done upon the system by outside agents and 
no work would be done by the system upon any outside body. 
Such an ideal system is called a closed system. The sun and 
planets constitute sensibly a closed system in so far as their 
mechanical actions and reactions are concerned, that is to say, 
the motion of the sun and planets is modified by the forces which 
they exert on each other but not perceptibly modified by the 
forces which are exerted upon them by the distant stars. 

Consider a closed system (the solar system for example) the 
particles* of which are in motion, and let us consider what takes 
place in any short interval of time. In the first place, each 
particle moves through a certain small distance, the configuration 
of the system is changed accordingly, and the potential energy of 
the system decreases (or increases) by an amount which is equal 
to the work done on all the particles by their mutual force actions. 
In the second place, the velocity of each particle is changed by 
the forces acting upon it, the kinetic energy of each particle 
increases (or decreases) by an amount equal to the work done 
upon it, and the total kinetic energy of the system increases (or 
decreases) by an amount which is equal to the work done on all 
the particles by their mutual force actions. Therefore the de- 
crease (or increase) of potential energy of a closed system is 
always equal to the accompanying increase (or decrease) of the 
kinetic energy of the system, or in other words, the sum of the 
potential and kinetic energies of a closed system is constant. 

59. The principle of the conservation of energy. It may ap- 
pear from the argument of the preceding article that the total 
energy of a closed system must always be constant. This indeed 

*The individual planets and even the sun may be considered as particles in so 
far as their action on each other is concerned because they are so far apart. 



FRICTION. WORK AND ENERGY. 



133 



is true if the idea of potential energy is a legitimate idea, that is to 
say, if the work done by the mutual force actions between the particles 
of the system when the system is changed from one configuration to 
another configuration is independent of the intermediate stages 
through which the system is made to pass, or in other words, if the 
system is what is called a conservative system. This constancy 
of the total energy of a closed system of the kind specified (a 
conservative system, indeed all systems are conservative so far 
as known) is called the principle of the conservation of energy and 
reduced to its simplest terms it is that the work done by a system 
depends only upon the initial and final states of the system and 
it is hopeless to seek a roundabout method for bringing the 
system back to its initial state by a smaller expenditure of work. 
The usual statement of the principle of the conservation of 
energy is that energy can neither be created nor destroyed; but this 
statement is so completely abstracted from actual physical con- 
siderations that it is almost meaningless. 

60. Application of the principle of work to statics. The prin- 
ciple of virtual work. Consider a body which is acted upon by 
a number of forces. If the body were to be given any slight dis- 




placement whatever a certain total amount of work (called virtual 
work) would be done by the forces. This virtual work is equal to 
zero when the forces are in equilibrium. Thus, Fig. 51a represents 
a body in equilibrium under the action of the three forces A , B 



134 



ELEMENTS OF MECHANICS. 



and C. If the body were moved a small distance upwards, a 
small amount of work would be done on the body by the force C, 
and an equal amount of work would be done by the body on the 
agents which exert the forces A and B, that is to say, the total 
work which would be done on the body by the forces A , B and C 
during the displacement would be equal to zero. 

Proof of the parallelogram of forces. Consider two forces A and B which act 
upon a particle O as shown in Fig. 51&. The resultant R of these forces may be 





A 

\ 

Fig. Sib. Fig. 51c. 

defined as that force which would do the same amount of work as the forces A and 
B together during any small displacement whatever of the point 0. Imagine the 
point O to be displaced a distance Ax along the %-axis of reference and let X a and 
Xb be the ^-components of the given forces A and B, respectively. The work 
done by the two forces A and B during the displacement Ax would be ( X a -\- Xb) ■ Ax. 
Let X r be the ^-component of the force R. The work done by R during the dis- 
placement A x would be Xr • Ax. Therefore placing these two expressions of virtual 
work equal to each other, we have 

Xr 'Ax=(Xa+Xb) 'Ax 

Xr=Xa+Xb (i) 

That is to say, the x-component of the resultant of two forces is the sum of the 
^-components of the two forces and a similar statement applies to y-components. 
Now the projection of the diagonal of any parallelogram on an axis of reference is 
equal to the sum of the projections of the two adjacent sides of the parallelogram 
on that axis. Therefore equation (i) means that the resultant of two forces is 
represented by the diagonal of a parallelogram of which the sides represent the 
given forces. This argument proves that the resultant of A and B is equal and 
parallel to R. To show that it is colinear with R, consider that the two given 
forces A and B have zero torque action about the point O so that the single force 
which is equivalent to A and B combined must have zero torque action about the 
point O and it must therefore pass through the point O. 

Definition of torque. The torque action of a force about a given axis may be 
defined as that factor by which a slight angular displacement about the axis must 
be multiplied to give the work done by the force during the displacement. Thus, 
Fig. 51c represents a body pivoted on an axis O and displaced through a small 
angle A0 about 0. The point of application of the force F moves through a dis- 



FRICTION. WORK AND ENERGY. 



135 



tance r • A0, and the work done by F is Fr • A0Xcos 6. That is, the factor by 
which the angular displacement A0 is multiplied to give the work done is 
Fr • cos Q, and this is the measure of the torque action of the force F about the 
axis 0. 

Example 1. Application of principle of virtual work to the 
lever. Let the body shown in Fig. 51a be turned about the point 
(the fulcrum) through the angle A<p thus causing the point 
of application of force A to move downwards through the distance 
a - Ap, and the point of application of force B to move upwards 
through the distance b-A<p; then Aa-A<p is the work done on 
the body by force A , and Bb • A<p is the work done by the body 
on the agent which exerts force B. Therefore Bb-Acp is to be 
considered as negative work done on the given body, so that the 
total work done on the body by the forces A and B during the 
given displacement is Aa-A<p + Bb-A<p and this is equal to 
zero if the forces are in equilibrium. Therefore we have 

Aa-Ay + Bb-A<p = o 
or 

Aa + Bb = o 

which is the well-known equation of the lever. 

Example 2. Application of the principle of virtual work to a 
barrel hoop. Figure 52 repre- 
sents a hoop which can be tight- 
ened by means of the bolt b. 
Imagine the bolt b to be short- 
ened by a certain very small 
amount I. The work done in 
thus shortening the bolt would 
be 77, where T is the force 
exerted by the bolt on either of 
the flanges, that is to say, T is 
the tension of the hoop ; but to 
shorten the bolt by the amount 
/ would shorten the radius of the 
hoop by the amount Z/271-, so that each unit of circumfer- 
ence of the hoop would move inwards through a distance 




Fig. 52. 



136 ELEMENTS OF MECHANICS. 

l/2ir. Let F be the outward force exerted by the barrel on each 
unit of length of the hoop (or the inward force exerted on the 
barrel by each unit of length of hoop), then F X l/2w would 
be the work done by each unit length of hoop during the shorten- 
ing of the bolt, and the total work done by the entire hoop 
would be F X l/2ir X 2irr } where r is the radius of the hoop, 
Placing this work done by the hoop equal to the work done on 

the bolt we have 

I 
Tl = F X — X 2irr 
2tt 

or fc . 

T = Fr 

in which T is the tension of a hoop, r is the radius of the hoop, 
and F is the force with which each unit of circumference of the 
hoop pushes inwards on the barrel. In this discussion the stiff- 
ness of the hoop is ignored. 

Problems. 

75. A 165-pound man climbs a height of 40 feet in 11 seconds. 
How much work is done, and at what rate? Express the work 
in foot-pounds, and in joules; and express the power in foot- 
pounds per second, in horse-power, and in watts. Ans. 6,600 
foot-pounds or 8,940 joules of work done at the rate of 600 
foot-pounds per second, or 814 watts, or 1.09 horse-power. 

76. A horse pulls upon a plow with a force of 100 pounds 
weight and travels 3 miles per hour. What power is developed? 
Express the result in foot-pounds per second, in watts, and in 
horse-power. Ans. 440 foot-pounds per second; 597 watts; 0.8 
horse- power. 

77. A belt traveling at a velocity of 70 feet per second trans- 
mits 360 horse-power. What is the difference in the tension of 
the belt between the tight and loose sides in pounds weight? 
Ans. 2,829 pounds-weight. 

Note. When a belt drives a pulley the tension of the belt is greater on one side 
than on the other. Let F be the tension of the belt on the tight side and F' the 
tension of the belt on the loose side, and let v be the velocity of the belt. The 



FRICTION. WORK AND ENERGY. 1 37 

rate at which work is done on the driven pulley by the tight side of the belt is Fv 
and the rate at which work is delivered back to the driving pulley by the loose 
side of the belt is F'v. Therefore the net rate at which work is delivered to the 
driven pulley is (F—F')v. 

78. A stream furnishing 500 cubic feet of water per second 
falls a distance of 15 feet. What power can be developed from 
this stream by a water wheel of which the efficiency is 6ofo ? 
Ans. 511 horse-power. 

79. The engines of a steamship develop 20,000 horse-power, 
of which 30 per cent, is represented in the forward thrust of the 
screw in propelling the ship at a speed of 17 miles per hour. 
What is the forward thrust of the screw in pounds- weight? Ans. 
132,350 pounds-weight. 

Note. The useful part of the power developed by the engines of a steamship 
is represented by the forward thrust of the propellor shaft against the framework 
of the ship, and the useful power is equal to the product of this force times the 
velocity of the ship. 

80. An electric motor has an efficiency of 80 per cent, and 
electrical energy costs 5 cents per kilowatt-hour. How much 
does the output of the motor cost per horse-power hour? Ans. 
4.66 cents. 

81. A 1,000 horse-power boiler and engine plant costs about 
$70,000 complete, including land, building, boilers, engines and 
auxiliary apparatus such as pumps and feed water heaters. The 
cost of operating this plant continuously, night and day, is as 
follows : 

Interest on investment .... 5 per cent, per annum. 

Depreciation . . . . . . 10 " " " " 

Maintenance and repairs . . . . 4 " " " " 

Taxes and insurance . . . . . 2 " " " " 

Labor $30 per day, 365 days in year. 

Coal $2.00 per ton. 

The average demand for power is 50 per cent, of the rated 
power output of the plant, that is 500 horse-power, and the con- 
sumption of coal is 2 ^2 pounds per horse-power-hour. Find the 
cost of a horse-power-hour delivered by the engine. Ans. 0.83 
cent. 



138 ELEMENTS OF MECHANICS. 

82. The above engine will drive a 700 kilowatt dynamo, that is 
a dynamo capable of delivering 700 kilowatts, The cost of 
dynamo, station wiring and switch-board apparatus is $20,000. 
The average output of the dynamo is 350 kilowatts (correspond- 
ing to 500 horse-power output of engine) . Calculate the cost of 
electrical energy per kilowatt-hour at the station, allowing 21 per 
cent, for interest, depreciation, etc., on the electrical machinery 
and allowing $5 per day additional for labor. Ans. 1.38 cents. 

83. A steam engine indicator shows an average steam pressure 
of 55 pounds per square inch (reckoned above atmospheric pres- 
sure) during each stroke of a steam engine, and the engine ex- 
hausts into a condenser where the pressure is 13 pounds per 
square inch below atmospheric pressure. The diameter of the 
piston is 16 inches, the diameter of the piston rod is 3 inches, the 
length of stroke is 24 inches, and the engine makes 75 revolutions 
per minute. Find the indicated horse-power of the engine. Ans. 
1 19.9 horse-power. 

Note. In solving this problem consider that the piston rod projects out of both 
ends of the cylinder so that the steam acts upon the piston surface which is outside 
of the piston rod on both strokes. 

84. A brake test of a steam engine gave the following data: 
speed of engine 200 revolutions per minute, length of brake arm 
(a, Fig. 49) 7% feet, observed force at end of brake arm and at 
right angles to arm 240 pounds- weight. Find the brake horse- 
power of the engine. Ans. 68.6 horse-power. 

Note. Reduce the force at the end of the brake arm to the equivalent force 
at the circumference of the pulley, and consider that the rim of the pulley is being 
moved in opposition to this force at a known velocity, so that the work done in 
driving the pulley is equal to the product of the force times the velocity of the rim 
of the pulley. 

85. A fan blower is mounted on a cradle which swings on knife 
edges in the line of the axis of the fan. When the fan is driven, 
the cradle tends to tip to one side and this tendency is balanced 
by a weight sliding on a horizontal lever arm, as shown in Fig. 
85^. The belt is thrown off the fan, the sliding weight moved to 
give a balance and the "zero position" of the weight is observed. 



FRICTION. WORK AND ENERGY. 



139 



The fan is then driven at a speed of 1,800 revolutions per minute 
and the weight (10 pounds) has to be moved 6^ inches from its 
zero position to balance the driving torque exerted by the belt on 




Fig. 85£. 

the fan. Find the power expended in driving the fan and express 
it in horse-power and in watts. Ans. 1.93 horse-power; 1,440 
watts. 

Note. Let r be the radius of the pulley. Then the velocity of the belt is known 
in terms of r. Let F be the tension of the belt on the tight side and F' the tension 
of the belt on the loose side. Then (F—F')r is the total torque action exerted 
on the fan by the belt, and this torque is balanced by moving the 10-pound weight 
through a distance of 6% inches. Therefore from the equation 

(F _ p')r = — - X 10 pound-feet 
12 

the value of F—F' can be determined in terms of r. 

When the velocity of the rim of the pulley (the velocity of the belt) is multiplied 
by F—F' the unknown radius r cancels out, and the power delivered to the fan 
is completely determined. 

86. Four idle pulleys A, B, C and D, Fig. S6p, are mounted 
in a frame which is free to rotate about the point which is the 
point of intersection of the left hand stretches of belt. A weight 
W slides along a lever arm which is fixed to the rocking frame 
so that the tilting action of the right hand stretches of belt may 
be balanced and measured. When no power is transmitted by 
the belt, the tension of the belt is the same everywhere and, under 
these conditions, the weight W is adjusted to its zero position to 
give a balance. When power is transmitted to a given machine the 
belt tensions F' and ^"differ and the weight W is moved farther 



140 



ELEMENTS OF MECHANICS. 




Fig. S6p. 

away from to give a balance. This movement of the weight W 
from its zero position is 16 inches, the mass of W is 55 pounds, 
the distance r is 24 inches, and the speed of the belt is 80 feet per 
second. Find the power transmitted by the belt. Ans. 5.33 
horse-power. 

Note. To find the value of F'—F" consider the balance of torque actions 
about the point O. 

87. A shaft transmits 100 horse-power and runs at a speed of 
250 revolutions per minute. Calculate the torque exerted on the 
shaft. Express the result in pound-feet, in pound-inches, and in 
dyne-centimeters. Ans. 2,100 pound-feet; 25,200 pound-inches 
284 X 10 8 dyne-centimeters. 

Note. The simplest argument of this problem is to imagine the shaft to be cut, 
and then coupled by a device shown in Fig. 8jp. The required torque is equal 





Fig. 87p. 

to the product of the force F times the arm r, and the power which is transmitted 
is equal to the product of the force F times the velocity of the pin p 

88. A steamship has a gross mass of 25,000 tons. What is 
the kinetic energy of the ship at a speed of 1 8 miles per hour ? 



FRICTION. WORK AND ENERGY. 



141 



Express the result in foot-pounds and in horse-power-hours. 
Ans. 544.5 X 1 o 6 foot-pounds; 275 horse-power-hours. 

89. A bicycle rider has a 50-foot hill to climb. What velocity 
must he have at starting to relieve him from the doing of one 
third of the work required? Ans. 32.7 feet per second. 

90. The rim of the fly-wheel of a metal punch is 5 feet in 
diameter and its mass is 560 pounds. At what initial speed must 
the fly-wheel run in order that the punch may exert a force of 
72,000 pounds through a distance of one inch and reduce the 
speed of the fly-wheel only 30 per cent? Ans. 2.33 revolutions 
per second. 

91. A counterpoise of % pound balances a weight of 100 
pounds wherever the weight may be placed on the platform of 
a balance scale. In what way and to what extent does the plat- 
form move when the counterpoise moves % inch downwards? 
Ans. Every part of the platform moves 0.00375 inch upwards. 

Note. This problem may be solved in the simplest possible manner by using 
the principle of virtual work. 

92. A screw-jack is turned by a lever of which the radial 

length is 18 inches, and the pitch of the screw is %& inch. What 

is the lifting force produced by a pull of 100 pounds 

on the end of the lever, neglecting friction? Ans. 

30,160 pounds-weight. 

Note. Consider the distance travelled by the end of the lever 
and the travel of the screw in one complete turn, and apply the 
principle of virtual work. 

93. The differential pulley consists of a large pulley 
A and a smaller pulley B made in one piece, and 
a third pulley C, all threaded with an endless chain 
as shown in Fig. 93^. The pulleys A and B are 
sprocket wheels with notches which engage the links 
of the chain so that the chain cannot slip on A and 
B. One turn of A and B takes in at a a length of Ficr p 
chain which is equal to the circumference of the larger 

pulley A and pays out at b a length of chain which is equal 
to the circumference of the smaller pulley B. 



142 ELEMENTS OF MECHANICS. 

The circumference of A contains 12 notches, the circumference 
of B contains 1 1 notches and the length of each link of the chain 
is 1 % inches. What is the lifting force produced by a pull of 
150 pounds at F, neglecting friction? Ans. 3,600 pounds- weight. 

Note. When the point of application of the force F in Fig. g^p moves downward 
through a distance x the weight W is lifted through a distance y. The values of 
x and y may be easily found from the data of the problem. 

94. A wooden tank 10 feet in diameter is held together by 
hoops of steel. The tension in one of the hoops is 1,000 pounds 
of force. Find the force exerted against the tank by each foot- 
length of hoop. Ans. 200 pounds of force: . 



CHAPTER VI. 

ROTATORY MOTION. 

61. Rotation about a fixed axis. Definitions. The simplest 
case of rotatory motion is that which is exemplified by the rota- 
tion of a wheel about a fixed axis. We shall first consider this 
simple case in detail and then proceed to the more complicated 
rotation about a moving axis. In order to rivet the attention to 
rotatory motion to the exclusion of movements of distortion, the 
idea of a rigid body will be used throughout the chapter, a rigid 
body being an ideal body which cannot change its shape or size. 

Angular velocity. Let <p be the angle turned by a rotating 
body during / seconds; the quotient <pft is called the average 
angular velocity of the body during the t seconds. If the time 
interval is very short, the quotient A<p/At is the actual angular 
velocity of the body at the given instant, A<p being the angle 
turned by the rotating body during the short time interval At. 
When the angle <p is expressed in radians and time / in seconds, 
then the quotient <p/t is in radians per second. Angular velocity 
is expressed in radians per second throughout this chapter. In 
practice, angular velocity is generally expressed in revolutions per 
second. There are 2ir radians in one revolution, and therefore 
one revolution per second is equal to 2w radians per second, or, 
in general 

co = 2irn (28) 

in which co is the angular velocity of a body in radians per second 
and n is the angular velocity in revolutions per second. 

Angular velocity is frequently called spin-velocity or, simply, 
spin. 

Angular acceleration. In many machines a part may rotate at 
a variable angular velocity. This is most strikingly illustrated 

143 



144 



ELEMENTS OF MECHANICS. 



by the motion of the balance wheel of a watch. The rate of 
change of the angular velocity of a body is called its angular 
acceleration. Thus an engine is started, and after six seconds the 
fly wheel has an angular velocity of 4 revolutions per second 
(= 25.13 radians per second), so that the average angular accel- 
eration of the wheel during the six seconds is 4.1888 radians per 
second per second. Of course, the fly wheel may have gained 
most of its angular velocity during a portion of the six seconds, 
so that 4.1888 radians per second per second is merely its aver- 
age angular acceleration. The angular acceleration of a rotating 
body at a given instant is equal to the quotient Aw/At where 
Aw is the angular velocity gained during the short interval of 
time At. 

Angular acceleration is frequently called spin-acceleration. 

62. Unbalanced torque and angular acceleration. Definition 
of moment of inertia. When a wheel is set in rotation, an unbal- 
anced torque must act upon the wheel. This is exemplified in 
the operation of spinning a top. When a rotating wheel is left 
to itself it loses its angular velocity and comes to rest on account 
of the friction of the wheel against the air and on account of the 




wheel 




Fig. 53a. Fig. 536. 

friction of the shaft in its bearings. To maintain a steady motion 
of rotation of a wheel, a driving torque must act upon the wheel 
sufficient to balance the opposing torque due to friction. 



ROTATORY MOTION. 1 45 

The effect of an unbalanced torque is to change the angular 
velocity of a wheel, or, in other words, or produce angular 
acceleration, positive or negative as the case may be. The 
angular acceleration of a given wheel is proportional to the un- 
balanced torque which acts upon the wheel, and a given unbal- 
anced torque produces a small angular acceleration of a large 
heavy wheel, or a large angular acceleration of a small light 
wheel. Thus, if a cord be wrapped around the shaft upon which 
a wheel is mounted, a pull on the cord produces torque equal to 
Fl, see Figs. 53a and 53^; and this torque imparts angular velocity 
to the small light wheel, Fig. 53a, at a rapid rate, whereas it 
imparts angular velocity to the large heavy wheel, Fig. 536, at a 
much slower rate. 

When a wheel is rotating every particle of the wheel moves at 
a definite linear velocity, and when the angular velocity of the 
wheel increases it is evident that the linear velocity of every 
particle of the wheel must increase; that is to say, angular 
acceleration of a wheel involves linear acceleration of every particle 
in the wheel, and it is possible to show the exact relation between 
angular acceleration and the unbalanced torque which produces 
it, by considering the linear ac- 
celeration of each particle in a 
wheel. The following discussion 
of this matter is the foundation of 
the dynamics of rotatory motion 
and it leads to a definition of 
what is called the moment of iner- 
tia of a wheel. 

Figure 54 represents a wheel 
rotating n revolutions per second, 

Fig. 54. 

or 2irn radians per second, about 

the axis 0. The particle Am describes a circular path of which 
the circumference is 2irr, the particle traces this circumference 
n times per second, and therefore the linear velocity v of the 
particle is 2wrn centimeters per second, r being expressed in 




I46 ELEMENTS OF MECHANICS. 

centimeters; but 2irn is equal to the angular velocity co of the 
wheel in radians per second, and, therefore 

v = ru (29) 

If the angular velocity of the wheel is changing, the linear 
velocity v of the particle m must change r times as fast as w, 
inasmuch as v is always r times as large as w. Therefore, repre- 
senting the angular acceleration of the wheel by a (rate of change 
of co) and representing the linear acceleration of the particle by 
a* (rate of change of v) we have 

a = ra (30) 

In order to produce the acceleration a of the particle, an un- 
balanced force F, see Fig. 54, must act on the particle in the 
direction of a, this force, expressed in dynamic units, must be 
equal to Am -a according to equation (1) Art. 33, and the torque 
action of this force is equal to Fr(= Am -a X r)\ but a is equal 
to ra according to equation (30), so that Fr = Am'r 2 a, or repre- 
senting Fr by AT, we have 

AT = ar 2 -Am 

in which A T is that part of the unbalanced torque T acting on 

the wheel which causes the linear acceleration of the given 

particle Am. Consider in this way all of the particles of the 

wheel and we have 

AT = ar 2 -Am 

A7\ = ar l 2 -Am l 

AT 2 = ar 2 -Am 2 

AT 3 = ar 2 -Am 3 

etc., etc., whence, by adding, we have 

*We are not concerned here with the radial acceleration of the particle Aw, since 
the radial acceleration is produced by unbalanced radial forces which have no torque 
action about 0. Radial accelerations of the particles of a wheel have nothing to do 
with the angular acceleration of the wheel. 



ROTATORY MOTION. 147 

T = a(r 2 -Am + r 2 -Am l + r 2 Am 2 -f • • •) 



or 

or writing 

we -have 



T = a~2r 2 -Am 



K = 2r 2 -Am (31) 

T = Ka (32) 



The quantity K, which is obtained by multiplying the mass of 
each particle of the wheel by the square of its distance from the 
axis and adding all of these products together, is called the moment 
of inertia of the wheel, and equation (32) shows that the unbal- 
anced torque acting on a wheel is equal to the product of the 
moment of inertia of the wheel and the angular acceleration of 
the wheel. 

Units involved in equations (31) and (32). If c.g.s. units are 
used throughout, then moment of inertia is expressed in grams 
X centimeters squared (gr. cm. 2 ), torque is expressed in dynes 
X centimeters and, of course, angular acceleration is expressed in 
radians per second per second. Equations (31) and (32) hold 
good, however, when moment of inertia is expressed in pounds 
X feet squared (lb. ft. 2 ), torque in poundals X feet and angular 
acceleration in radians per second per second. If torque is 
expressed in pounds- weight X feet, moment of inertia in pound- 
feet-squared and angular acceleration in radians per second per 
second, then equation (32) becomes 



F= Ka (33) 

32.2 VJJ/ 

approximately. 

Example of the calculation of moment of inertia. The moment of inertia of a 
homogeneous solid of regular form can be calculated by the methods of calculus. 
Consider, for example, a long slim rod of length L and mass M rotating about its 
middle point as shown in Fig. 55. The mass of the short portion dr is (M/ L)Xdr 
and its distance from O is r. Therefore, writing (M / L)Xdr for Am inequation 
(31) we have 

but the sum (integral) ^>r 2 dr between the limits r=+L/2 and r= — L/2 is equal 



148 



ELEMENTS OF MECHANICS. 



to 12 L , so that X = T 1 I MI 2 . The moments of inertia given in the following table 
were calculated in this way. 

Radius of gyration. The radius of 
gyration of a rotating body is the dis- 
tance p from the axis of rotation at 
which the entire mass M of the body 
might be concentrated without alter- 
ing the moment of inertia of the 
body. If the entire mass M were 
concentrated at distance p from the 
axis, the moment of inertia would be 
equal to Mp 2 , according to equation 
(31). That is 




Fig. 55. 



K = Mp' 



(34) 



or 



TABLE. 
Moments of inertia of some regular homogeneous solids. 



Axis of rotation passing through center of mass. 



Sphere of radius R and mass M 

Cylinder of radius R and mass M, axis of cylinder is the axis 
rotation 

Slim rod of length L and mass M, axis of rotation at right angles 
to rod 

Rectangular parallelopiped of length L and breadth B, axis of ro- 
tation at right angles to L and B 



Value of K. 

\MR} 

{MR 2 

j\ML 2 

T \M(L 2 +B 2 ) 



Using the values of K in the above table, this equation shows 
that the radius of gyration of a sphere is V f times the radius of 
the sphere, the radius of gyration of a cylinder rotating about 
its axis of figure is V x / 2 times the radius of the cylinder, and the 
radius of gyration of a long, slim rod rotating about an axis at 
right angles to the rod and passing through its center of mass is 
V-12 times the length of the rod. 



ROTATORY MOTION. 1 49 

If a rotating body be imagined to be divided into particles of 
equal mass, then the radius of gyration may be defined as the 
square-root-of-the-average-square of the distances of all the par- 
ticles from the axis. 

Angular momentum. The product of the moment of inertia 
of a rotating body and the angular velocity of the body is called 
the angular momentum or spin momentum of the body, that is to 
say, the spin momentum of a body is equal to Ka>, where K is 
the moment of inertia of the body about its axis of spin, and o> 
is its spin velocity in radians per second. The spin momentum 
of a body cannot change except by the action of an outside torque 
upon the body. This fact is known as the principle of the con- 
servation of angular momentum. The following experiment fur- 
nishes a very striking illustration of this fact. A person stands 
upon a stool which turns freely about a vertical axis (on ball 
bearings), and he is set slowly rotating with his arms extended. 
By bringing his arms down to his sides he decreases his moment 
of inertia very considerably and produces a very great increase 
in his angular velocity. The change of moment of inertia is 
very much greater if the person holds weights in his hands. 
The fact here described is also illustrated by the formation of a 
whirlpool when a slowly rotating liquid flows out through a hole 
in the bottom of a bowl. The movement of the fluid towards the 
axis of the bowl reduces the moment of inertia of the system, 
and, according to the principle of constancy of angular momen- 
tum, the angular velocity is very greatly increased.* 

63. Kinetic energy of a rotating body. A rotating wheel evi- 
dently stores kinetic energy because it can do work while being 
brought to rest. The kinetic energy of a rotating body is given 
by the equation 

w=y 2 KJ, (35) 

in which everything is expressed in c.g.s. units. The proof of 
this equation gives, perhaps, a clearer idea of the significance of 

♦This matter is discussed in Art. 123. 



150 



ELEMENTS OF MECHANICS. 



moment of inertia than the discussion of equation (32) given in 
the foregoing article. Consider the rotating wheel shown in Fig. 
54. The linear velocity of the particle Am is rot, according to 
equation (29), and therefore the kinetic energy of this particle is 



AW 



4Am-r 2 o) 2 



according to equation (26). 

Consider in this way all of the particles of the wheel and we 

have 

AW = %Am-rW 

AW Y = yiAm^-rfJ ' 
AW 2 = }£Am 2 -r 2 V 
AW 3 = }4Am B -r z 2 o3 2 

whence, by adding, we have 

W = y 2 ^r 2 -Am 

and by comparing this equation with equation (31) we have 
equation (35). 

64. Relation between moments of inertia about parallel axes. 

Let K be the moment of inertia of a body of mass M about a 




Fig. 56. 

given axis passing through the center of mass of the body, and 
let K r be the moment of inertia of the body about another axis 



ROTATORY MOTION. 15 I 

parallel to the first and distant d from the center of mass, then 
K' = K + d 2 M (36) 

Let 0, Fig. 56, be the center of mass of the body, chosen as 
the origin of co-ordinates, let K be the moment of inertia of the 
body about an axis through perpendicular to the plane of the 
paper, and let K' be the moment of inertia of the body about 
an axis through 0' also perpendicular to the plane of the paper. 
Consider a sample particle of the body Am, distant r from 
and distant / from 0' ', and of which the coordinates are x and y.. 
By trigonometry we have 

Y ' 2 = r 2 _j_ d 2 — 2rd cos (i) 

From equation (31) we have 

K' = 2r' 2 Am (ii) 

whence, substituting the value of r' from equation (i) we have 

K' = 2Zr 2 -Am + 2d 2 Arn — 2dZr cos 6- Am (iii) 

but Sr 2 . Am is equal to K, and 2Zd 2 Am is equal to d 2 M. Fur- 
thermore 2> cos 6 . Am is equal to Xx . Am, which is equal to 
zero according to equation (22) since the origin of coordinates 
is chosen at the center of mass of the body. Therefore equation 
(iii) reduces to equation (36). 

65. Equivalent mass of a rolling wheel. If a wheel and axle 
is set moving by an applied force F as shown in Fig. 57a, a string 
being wrapped around the axle and fastened to a rail as indicated 




Fig. 57a. 

in the figure, then a backward force G will be exerted on the 
wheel and axle by the string, and the net unbalanced force which 
is producing forward acceleration of the wheel will be F — G. 
The same conditions exist when a wheel is set rolling along a 



152 ELEMENTS OF MECHANICS. 

track, that is to say, while the wheel is gaining forward velocity 
the track exerts a backward force at the rim of the wheel like 
the force G in Fig. 57a. Also, while a railway car is gaining 
forward velocity, the track exerts a backward force at the rim 
of each wheel like the force G in Fig. 57a. This is especially 
true in the case of a trolley car * which has rapidly rotating motor 
armatures geared to the car axles. Therefore, a given force 
applied to a car produces less forward acceleration than it would 
produce if the car were to slide along a frictionless track instead 
of moving forwards on rolling wheels. 

When it is desired to take this effect into- account in the dis- 
cussion of the motion of a car in practice, it is usual to assign 
to the car a fictitious mass in excess of its actual mass so that the 
forward acceleration produced by an applied force F acting alone 
would be the same as the forward acceleration produced by the 
two forces F and G on the actual car. This fictitious mass is 
called the equivalent mass of the car and wheels. 

It is sufficient for present purposes to discuss the equivalent 
mass of a rolling wheel by itself. The equivalent mass of the 
wheel is most easily defined as that mass M which would store 
the whole kinetic energy of the rolling wheel at a forward velocity 
equal to the linear velocity v of the wheel. Thus we may write 

}4Mv 2 = j4mv 2 + j4 Ka> 2 (i) 

in which m is the actual mass of the wheel, K is the moment of 
inertia of the wheel, and co is the angular velocity of the wheel. 
The angular velocity co, however, satisfies the eqaution 

v = r (a (29)bis 

and therefore, substituting v/r for co in equation (i) we have 

\Mv 2 = imv 2 + if 2 v 2 
or 

M = m + ^ (37) 

*This statement refers to an ordinary trolley car with idle motors, the car being 
hauled along by another car or coasting down hill. 



ROTATORY MOTION. 



153 



Examples, (a) The rolling motion of the wheels of a railway 
train causes the train to behave, in so far as the relation between 
linear acceleration and draw-bar pull of locomotive are con- 
cerned, as if its mass were greater than its actual mass by the 




Fig. sib. 

amount n K/r 2 , where n is the number of wheels, r is the diameter 
of the rolling circle of each wheel, and K is the moment of inertia 
of each wheel. In the case of an electric car with a geared motor, 
the moment of inertia of the motor armature can be reduced to 
an equivalent moment of inertia of wheel and thus be included 
in the value of K in equation (37), by multiplying the moment 
of inertia of the motor armature by the square of the gear ratio 
(ratio of the diameters of the rolling circles of the two gears). 
(b) Consider a metal sphere of mass m and radius r rolling 
down an inclined plane as shown in Fig. 576. The vertical pull 




Fig. 58. 

of the earth mg has a component parallel to the plane which is 
equal to mg sin 6, and this force would cause the ball to move 
with an acceleration equal to g sin if the inclined plane were 



154 ELEMENTS OF MECHANICS. 

frictionless so that the sphere would slide; but on account of 
the rolling motion the sphere behaves as if its mass were -J times 
m, according to equation (37), and therefore it rolls down the 
plane with an acceleration of only f of g sin 0. 

A wheel and axle rolling on a track as shown in Fig. 58 has a 
rolling circle of small radius r , so that its equivalent mass is very 
large, according to equation (37), and, therefore, such a wheel and 
axle rolls down an inclined plane with a very small acceleration. 

66. Correspondence between translatory motion and rotatory 
motion. To every equation in translatory motion there is a 
corresponding equation in rotatory motion in which moment of 
inertia K takes the place of mass m, angle takes the place of 
distance, angular velocity co takes the place of linear velocity v, 
angular acceleration a takes the place of linear acceleration a, 
and so on. The following table exhibits the pairs of correspond- 
ing equations. 







TABLE. 




Translatory 


motion. 




Rotatory motion. 


F = ma 


(3) 




T= Ka (32) 


W=Fd 


(24) 




W= T<$> (i) 


P = Fv 


(25) 




P=Tw (ii) 


W= l /2mv 2 


(26). 




W = Y 2 Ko>* (35) 


F=-kx 


(15) 




T=-kcP (38) 


k = 47r 2 n :? m 2 


(16) 




K = 4ir 2 n 2 K (39) 



Of these equations, those numbered (i), (ii), (38) and (39) have 
not been previously discussed; equation (i) refers to the work 
W done by the torque T in turning a body through an angle <p, 
axis of torque and axis of motion being parallel ; and equation 
(ii) refers to the power P developed by a torque T which acts on 
a body rotating at angular velocity co, axis of torque and axis of 
motion being parallel. 

Equations (38) and (39) refer to harmonic rotatory motion, 
that is, to oscillatory motion about an axis, such as is exemplified 
by the motion of the balance wheel of a watch. 

The equations of circular translatory motion correspond to the 
equations of the gyroscope to a limited extent as explained in 
Art. 72. 



ROTATORY MOTION. 155 

67. Rotatory harmonic motion. Consider a weight suspended 
by a steel wire. The weight will stand in equilibrium with the 
wire untwisted. If the weight is turned around the wire as an 
axis through the angle <p from this equilibrium position, then the 
twisted wire will exert a torque T on the weight tending to turn 
it back, and this torque will be proportional to <p, that is 

T = - k<p (38) 

in which the factor k is a constant for a given wire; it is called 
the constant of torsion of the wire. 

A weight suspended by a steel wire oscillates back and forth 
when the weight is turned about the wire as an axis and released, 
and the oscillatory motion of the weight constitutes what is 
called harmonic rotatory motion. Equation (38) is exactly similar 
in form to equation (15) (see table in Art. 66), and therefore the 
number n of oscillations of the weight per second and the moment 
of inertia K of the weight satisfy equation (39) which is exactly 
similar in form to equation (16). Therefore 

k = 4<ir 2 n 2 K (39) 

or, using i/r for n, where 7 is the period of one oscillation, we 
have 

K = — 2 — (40) 

T 

A weight hung by a wire and set oscillating about the wire as 
an axis, is called a torsion pendulum. 

68. Use of the torsion pendulum for the comparison of mo- 
ments of inertia. The constant of torsion, k, equations (39) and 
(40) , is nearly independent of the amount of weight supported by 
the wire, if the weight is not excessive, therefore, if two bodies 
are hung from the same wire, one at a time, and their respec- 
tive periods of torsional vibration r and r' observed, then from 
equation (40) we have 

4tt 2 K 

K = — 2- (l) 

T 

and 



156 



ELEMENTS OF MECHANICS. 



2F/ 



K = 



4ir z K 



whence 



K_ 
K' 



(ii) 
(iii) 



from which K' may be calculated if K is known. For example, 
one of the suspended bodies may be a homogeneous circular disk 
of which the moment of inertia is known (see table in Art. 62). 

69. The gravity pendulum consists of a rigid body AB, Fig. 
59, suspended so as to be free to turn about a horizontal axis 0. 

Let C, Fig. 59, be the center of mass of 
the body. This point C is vertically 
below when the body is in equilib- 
rium. Let the body be swung to one 
side through the angle <p, as shown. 
Then the force, Mg, with which the 
earth pulls the body tends to swing the 
body back to the vertical position with a 
torque T which is equal to the product 
of Mg and the length of the arm a C . 
But the distance a C is equal to x sin 
<p, where x is the distance OC. There- 
fore 

T = Mgx sin <p (i) 

When <p is small, then sin <p = <p, very 
nearly, and equation (i) becomes 




Fig. 59. 



T = Mgx-v* (ii) 

Comparing this with equations (38) and (40), we find that 

4t 2 K 



= Mgx 



(41) 



*Of course this equation may be written T=-Mgx ■ <t> because T tends to 
reduce 4>. 



ROTATORY MOTION. 157 

in which K is the moment of inertia of the body about the axis 
0, t is the period of one complete pendulous vibration of the 
body, and g is the acceleration of gravity. 

A pendulum such as here described is sometimes called a 
physical pendulum to avoid confusion with the ideal simple pendu- 
lum described in Art. 43. 

The simple pendulum. An ideal pendulum consisting of a 
particle of mass M suspended by a weightless cord, or rod, of 
length / is called a simple pendulum. The moment of inertia of 
such a pendulum about the supporting axis is K = Ml 2 , ac- 
cording to equation (31). Furthermore, the center of mass of a 
simple pendulum is, of course, at the center of the suspended 
particle. Therefore, for the simple pendulum, we may write 
Ml 2 for K, and / for x in equation (41), whence we have 

^2-=g (42) 

or 

T 2 g 
1 = ^? (43) 

4?r 

in which I is the length of a simple pendulum, r is the period of 
one complete vibration of the pendulum and g is the acceleration 
of gravity. 

Equivalent length of a physical pendulum. The length of a 
simple pendulum which would have the same period of vibration 
as a given physical pendulum is called the equivalent length of 
the given physical pendulum. Now, according to equation (43), 
the length of a simple pendulum, of which the period of one 
vibration would be t, is I = T 2 g\\Tr 2 . Therefore, solving equation 
(41) for r 2 gj \ir 2 (= T) we have 

1 = Mx ( 44) 

in which / is the equivalent length of a given physical pendulum, 
K is the moment of inertia of the pendulum about its axis of 



158 ELEMENTS OF MECHANICS. 

support, M is the mass of the pendulum, and x is the distance 
from the point of support to the center of the mass of the pen- 
dulum. 

The point in the line OC, Fig. 59, which is at a distance 
/(= K/Mx) from is called the center of oscillation of the 
pendulum. This point is also called the center of percussion of 
the pendulum for the reason that if the pendulum is started or 
stopped by a horizontal hammer-blow at this point no side force 
is produced on the supporting axis. See Art. 71. 

70. The determination of gravity. The, most accurate deter- 
mination of the acceleration of gravity is made by means of 
the pendulum. This determination would be a very 
simple thing if it were feasible to construct a simple pen- 
dulum, in which case equation (42) could be used for 
calculating gravity from the measured length, I, of the 
simple pendulum and its observed period r. The de- 
termination of the acceleration of gravity by means of 
an actual pendulum depends, however, upon the deter- 
mination of the moment of inertia of the pendulum, as is 
evident from equation (41), and the moment of inertia 
of a body cannot be determined with great accuracy. 
This difficulty is obviated by means of the so-called 
reversion pendulum which was devised by Henry Kater 
in 1818. 

A simple form of Kater's pendulum is shown in Fig, 
60. A stiff metal bar has two knife-edges, from either 
of which it may be swung as a pendulum, and two 
weights, WW, which may be adjusted until the period 
Fig 60 T °^ one vibration of the pendulum is the same- 
whether it be swung from a or b. Then the dis- 
tance between the knife-edges a and b is the equivalent length 
of the pendulum and may be used for / in equation (43).* 

Comparison of the values of gravity at two places by means of 
the pendulum. If the same pendulum be swung at two places in 

*See equation (46), below. 



ROTATORY MOTION. 159 

succession and its respective periods t and / observed, we have 
from equation (41) 

^f = M gX (i) 

r 

and 

^ = M g >x (ii) 

T 

in which g and g' are the respective values of the acceleration 
of gravity at the two places. Dividing equation (i) by equation 
(ii), member by member, we have 

From this equation the value of g may be accurately deter- 
mined at any place in terms of its known value at another place, 
by observing the values of r and r' of an ordinary pendulum, 
every precaution being taken to avoid variations of dimensions 
of the pendulum due to temperature or to careless handling. 
Most of the gravity determinations of the United States Coast 
and Geodetic Survey are made in this way, the value of g at 
Washington having been once for all determined with the greatest 
possible accuracy by means of Kater's pendulum. 

The accompanying table gives the value of g in centimeters per 
second per second at several places as determined by the pen- 
dulum. 

Theory of the reversion pendulum. — Consider a body of mass M, its center of 
mass at 0, Fig. 61. Let 0', 0, and 0" be co-linear points; let t' and r" be the 
vibration periods of the body swung as a pendulum from 0' and 0" respectively; 
and let K, K', and K" be the moments of inertia of the body about 0, 0', and 0" 
respectively. From equation (41) we have 



and 



From equation (36) we have 



T' 



Mgx (i) 



= Mgy (ii) 



K' = K + x 2 M (iii) 

K" = K + y 2 M (ivj 



i6o 



ELEMENTS OF MECHANICS. 



TABLE. 



Locality. 



Boston, Mass. . . . 
Philadelphia, Pa. . 
Washington, D.C. 

Ithaca, N. Y 

Cleveland, O 

Cincinnati, O. ... 
Terre Haute, Ind.. 

Chicago, 111 

St. Louis, Mo. . . . 
Kansas City, Mo. 

Denver, Col 

San Francisco, Cal. 



Latitude. 



42° 2l' 33" 

39 57 06 

38 53 20 
42 27 04 
41 30 22 

39 08 20 
39 28 42 
41 47 25 

38 38 03 

39 05 50 
39 40 36 
37 47 00 



Greenwich 51 29 00 

Paris 48 50 11 

Berlin 52 30 16 

Vienna j 48 12 35 

Rome 41 53 53 

Hammerfest I 70 40 00 



Longitude. 



Elevation. 



71 03' SO" 

75 11 40 
77 01 32 

76 29 00 
81 36 38 
84 25 20 
87 23 49 
87 36 03 
90 12 13 
94 35 21 

104 56 55 

122 26 00 



o 00 00 

2 20 15 

13 23 44 

16 22 55 

12 28 45 

22 38 00 



22 meters. 

16 

10 
247 
210 
245 
151 
182 

154 

278 

1638 

114* 

47 
72 

35 

150 

53 



Value of g 
(not Reduced 
to Sea-level). 



980.382 
980.182 
980.IOO 
980.286 
980.227 
979.990 
980.058 
980.264 
979.987 
979.976 
979-595 
979-951 

981.170 
980.960 
981.240 
980.852 
980.312 
982. 5S0 



Substituting these values of K' and K" in (i) and (ii), we have 

4 tt 2 (K + x 2 M) 

■ 5 ■ = Mgx 



(v) 



4->r 2 (K + y 2 M) 



Mgy 



(vi) 



f 



Eliminating K/M from (v) and (vi), we have 

4 W* 2 -y 2 ) 

■ o = S 

XT '2 _ y T »* 



(45) 



If t' = t", we may cancel (x — y), provided (x—y) is not equal 
to zero, giving 



47r 2 (x + y) 



= g 



(46) 



(1) If the pendulum has been adjusted by repeated trial, so 
that t' = t", then equation (46) enables the calculation of g, when 
(x+y) and t' have been observed. 

(2) If the pendulum has not been adjusted, equation (45) enables 
the calculation of g, when x, y, r', and r" have been observed. 

(3) If the pendulum has been roughly adjusted, so that r' 
Fig. 61. and r" are nearly equal, then equal and opposite errors in x and y 

very nearly annul each other in their influence upon the value of g 
as calculated by equation (45). Therefore equation (45) gives g very accurately 
when t' and t" are nearly equal, {x-{-y) being measured with great accuracy, and 



ROTATORY MOTION. l6l 

x measured roughly. The value of y is taken from (x-\-y) — x, so that its error 
may counteract the error due to the roughly measured value of x. The position 
of the center of mass 0, Fig. 6i, is found with sufficient accuracy for the rough 
measurement of x by balancing the pendulum hoiizontally on a knife edge. 

Note. When x = y, equation (46) is not necessarily true, since it has been 
derived from equation (45) by cancelling (x—y), which is zero. 

71. Motion of a rigid body when struck with a hammer. 

When an unbalanced force continues to act upon a body for an 
appreciable length of time, the problem of determining the motion 
of the body is complicated by the fact that, as the body moves, 
the force generally changes its point of application, or its value, 
or its direction, or all three of these things may change simulta- 
neously. The force due to a hammer blow, however, is of such 
short duration that the actual movement of the body during the 
time that the force acts is negligible, and the problem of finding 
the motion produced by the hammer blow is quite simple. A 
hammer blow is called an impulse and it is measured by the 
product of the average value, F, of the force exerted by the 
hammer and the short time t that the force continues to act. 
The impulse of a hammer blow when the hammer is brought to 
rest by the blow is equal to the momentum mv of the hammer. 
This is evident from the following considerations : Let F be the 
average value of the force acting to stop the hammer. Then 
F = ma, where m is the mass of the hammer and a is the average 
rate at which it loses velocity while stopping. Multiplying both 
members of this equation by the time which elapses during the 
stopping of the hammer, we have 

Ft = mat 

but the average rate a at which the hammer loses velocity 
multiplied by the time t is the total initial velocity of the hammer, 
and therefore the average force F exerted by the hammer while 
it is stopping multiplied by the time / occupied in stopping is 
equal to the momentum mv of the hammer. When m is ex- 
pressed in grams and v in centimeters per second, then mv is 
expressed in dyne-seconds. When m is expressed in pounds 
12 



l62 



ELEMENTS OF MECHANICS. 



(mass) and v in feet per second then mv is expressed in poundal- 
seconds. 

A rigid stick, AB, Fig. 62a, is struck with a hammer in the 
direction of the arrow, h, at a point distant x above the center 
of mass, 0, of the stick. The motion imparted to the stick by 
the blow is a combination of translatory motion and rotatory 
motion, but the combination of a constant translatory motion 
and a constant rotatory motion is exactly the kind of motion 
which is performed by a rolling wheel, and therefore the hammer 
blow causes the stick to' move as if the stick were attached to a 
weightless circular hoop, CC, and this hoop allowed to roll 
without friction on a straight rail. The center of the rolling 
circle, CC, is at the center of mass of the stick, and the radius, 
y, of the rolling circle depends upon the distance, x, and upon the 






\c 



W 



\ 



— t. rolling circle \ 



i °J 

y / 

1 / 

1 



/ 



Jtrl 



rail 



Fig. 62a. 



ratio of the moment of inertia of the stick (about 0) to the mass 
of the stick, according to equation (47). At the instant of the 
hammer blow the motion of the stick is equivalent to a simple 
motion of rotation about the point 0". 

To analyze the effect of the hammer blow, the translatory mo- 
tion and the rotatory motion may be treated separately. Re- 



ROTATORY MOTION. 1 63 

garding the translatory motion, we know from, Art. 48, that the 
velocity imparted to the center of mass is the same as if the whole 
mass of the stick were concentrated there and acted upon directly 
by the total force of the hammer. Let F be the average force due 
to the hammer, and / the time (very short) that it continues to act. 
Then F/M is the acceleration of the center of mass of the stick, 
and F/M multiplied by t is the velocity imparted to the center 
of mass, M being the mass of the stick. 

As to the rotatory motion of the stick, it is evident that Fx is 
the torque about due to the force of the hammer, so that Fx/ K 
is the angular acceleration of the stick during the time /, and 
Fx/ K multiplied by t is the angular velocity imparted to the 
stick by the hammer blow, K being the moment of inertia of 
the stick about 0. 

Now the whole stick is moving to the right at a velocity Ft/M 
on account of the translatory motion, and any point at a distance 
r below is moving to the left at a linear velocity equal to r 
times the angular velocity, Ftx/ K\ therefore, for the point 0" 
which is for the moment stationary, we must have, writing y for r, 

Ftxy _Ft 

K ~M 
or 

K 1 ^ 

which determines the radius y of the rolling circle when x and 
K/M are given. 

The problem of the base-ball bat. At the instant that a base-ball 
bat strikes a ball, the motion of the bat is a simple motion of 
rotation about a certain point 0" Fig. 62b; and, if the distances 
x and y satisfy equation (47), then the effect of the impact of 
bat and ball is to reduce the angular velocity of the bat about 
the point 0" without moving the point 0" '. The point of a bat 
which must strike a ball so that the impact may have no tendency 
to change the position of the point about which the bat is rotating 
at the instant of impact, is called the center of percussion of the 



1 64 



ELEMENTS OF MECHANICS. 



bat. The position of the center of percussion depends of course 
upon the position of the point 0" about which the bat is rotating 
at the instant of impact. 

72. Precessional rotatory motion. The foregoing articles refer 
to rotation about a fixed axis, or, as in the case of a rolling 
wheel, to rotation about an axis which performs translatory 
motion. The axis of a rotating body may, however, change its 
direction continuously. We shall discuss here only the com- 
paratively simple case * in which a symmetrical body spins about 
its axis of symmetry while at the same time the axis of spin 
rotates uniformly. This rotation of the axis of spin is called 





Fig. 62b. 



Fig. 63. 



precession, and the axis about which the axis of spin rotates is 
called the axis of precession. 

The gyroscope consists of a heavy wheel mounted on an axle 
which is pivoted in a metal supporting ring, as shown in Fig. 63 . 

*The student is referred to Poinsot's Theorie Nouvelle de la Rotation des Corps, 
which is perhaps the most intelligible account of the motion of a non-symmetrical 
rigid body. Spinning Tops and Gyroscopic Motion by H. Crabtree (Longmans, 
Green & Co., 1909) is perhaps the best simple treatise on this subject. The most 
complete treatise on the motion of a rigid body is Advanced Rigid Dynamics by 
E. J. Routh, •Macmillan & Company, London, 1892. 



ROTATORY MOTION. 



165 



The wheel is set in rapid rotation by wrapping a cord on the axle 
and giving the cord a vigorous pull. When the wheel is thus set 
rotating, the direction of the axle remains unaltered as long as 
no external twisting force, or torque, acts upon it; an unbalanced 
torque is necessary to change the direction of the axis of a rotating 
body, just as an unbalanced force is required to change the direction 
of translator y motion of a particle. 

In order to describe precisely how an unbalanced torque 
changes the direction of the axis of a rotating body, it is very 
convenient to represent angular velocity and torque by lines in a 
diagram. To represent an angular velocity by a line, draw the 
line in the direction of the axis of spin and of such length as to 
represent to scale the value of the angular velocity in radians 
per second; to represent a torque by a line, draw the line in the 
direction of the axis of the torque and of such length as to 
represent to scale the value of the torque in dyne-centimeters. 
In each case an arrow-head is to be placed on that end of the 
line towards which a right-handed screw would travel if turned 
in the direction of rotation in the one case, or if turned in the 
direction of the torque in the other case. 





Fig. 64a. 



Fig. 64b. 



1 66 ELEMENTS OF MECHANICS. 

Figure 64a is a top view of the gyroscope ; the metal ring rests 
upon a supporting pivot underneath the ring at 0, and the line 
OP, Fig. 645, represents the angular velocity 00 of the spinning 
wheel at a given instant. The pull of the earth on the wheel and 
ring produces an unbalanced torque about the axis OT, Fig. 
64a. The effect of this unbalanced torque, during a short in- 
terval of time, is to impart to the wheel an additional angular 
velocity Aco about the axis OT, and the resultant* angular 
velocity is then about the axis OP' , Fig. 646; that is, the effect 
of the unbalanced torque T is to cause the axis of spin to sweep 
around in the direction of the arrow ft. 

This effect of an unbalanced torque upon a rapidly rotating 
body is also exemplified by the motion of a spinning top. Thus 
the line OP, Fig. 65, represents the angular velocity of a 
spinning top. The vertical pull of the earth, mg, produces an 



Fig. 65. 

unbalanced torque about 0, and the angular velocity produced 
by this unbalanced torque, by being added continuously to OP 
as a vector, causes the axis of spin OP to sweep round the vertical 
axis OV in the direction indicated by the arrow ft. 

The force required to constrain a particle to a circular orbit 
depends upon the mass of the particle and upon the linear accel- 
eration which is involved in the continual change of direction of 

*The vector addition of angular velocities is explained in Art. 70. 



ROTATORY MOTION. 



167 



the velocity of the particle. The torque required to produce pre- 
cession of a spinning body depends upon the moment of inertia 
of the body and upon the angular acceleration which is involved 
in the continual change of direction of the axis of spin. Preces- 
sional motion of a spinning body corresponds to translatory 
motion in a circle. 

The torque required to produce precessional rotatory motion 
is given by the following equations : 



T = uVK 



(i) : 



when the axis of spin is at right angles to the axis of precession as 

in Figs. 63 and 64, or 

T = uQK sin <p (ii)* 

when the axis of spin makes an angle p with the axis of precession 
as in Fig. 65; in these equations co is the angular velocity of 
spin in radians per second, Q, is the angular velocity of precession 

\ 

\ 




\iQ, 



Fig. 66. 

*These equations are rigorously correct only when the rotating body is symmet- 
rical about the axis of spin and after steady precessional motion has been established, 
and equation (ii) is rigorously correct only when the rotating body is symmetrical 
about the point in Fig. 65 as it would be if it were a rotating sphere with its 
center at O. Equation (ii) is approximately true in a case like that shown in Fig. 65 
when the angular velocity of spin is very much greater than the angular velocity of 
precession. The precessional motion of the top in Fig. 65 about OV tends to in- 
crease the angle independently of the torque due to the force mg, and therefore 
the precessional motion of OP is more rapid than would be produced by the force 
mg alone if the top were symmetrical about the point O. 

The student is referred to a very excellent treatise on Spinning Tops by Harold 
Crabtree, for a more complete discussion of this subject and especially for the dis- 
cussion of the oscillations of a gyroscope or top (precessional motion not steady). 



168 ELEMENTS OF MECHANICS. 

in radians per second and K is the moment of inertia of the 
body about the axis of spin. 

Equation (i) may be established as follows : Let the line OA , 
Fig. 66, represent the angular velocity of spin of the body at a 
given instant, and let the dotted arrow $2 represent the angular 
velocity of precession. The angle turned by the axis of spin 
in a short interval of time At is = Q • At, and the increment 
of angular velocity which is represented by the line AB is 
equal to 0co, being very small. Therefore 

Ao> = uti-At 

whence 

Am 

77 = coft 

At 

but Aco/A/ is the angular acceleration of the rotating body, and 
the torque T which must act upon the spinning body to produce 
this angular acceleration is equal to the product of the angular 
acceleration and the moment of inertia of the body according to 
equation (32). Therefore we have 

T = uVK 

Equation (ii) may be established in the same way by consider- 
ing the component VP of the angular velocity of spin OP in 
Fig. 65. 

The above analysis of the action of the gyroscope will hardly be convincing to 
the beginner on account of the fact that the action is analyzed in terms of the rather 
complicated and unfamiliar ideas, angular velocity, angular acceleration, moment of 
inertia and torque; it is, therefore, desirable to analyze the action of the gyroscope 
in terms of the fundamental ideas of linear velocity and acceleration, mass, and 
linear force. The analysis of the action of the gyroscope in terms of linear velocity 
and acceleration depends upon a relation which is sometimes called Coriolis' law. 
Given a straight tube AB, Fig. 67, which is rotating about the axis C at angular 
velocity ^ as indicated in the figure. In this tube is a ball m which is moving away 
from C at velocity v (if the ball were moving towards C its velocity v would be con- 
sidered as negative). Under these assumed conditions the sidewise acceleration, a, 
of the ball m is equal to 2&v, that is 

a = 2&v (v) 

To derive this relation, the sidewise acceleration a may be considered in two parts. 
In the first place we have the acceleration which is associated with the continual 



ROTATORY MOTION. 



169 



change of direction of the radial velocity v of the ball. This acceleration is equal to 
tiv as shown in Fig. 68, and as explained in Arts. 18 and 38. In the second place, 
as the ball gets farther and farther away from the axis C, its actual sidewise velocity, 
due to the rotation of the tube, increases, but this sidewise velocity is equal to &r, 
according to equation (29), and therefore, since v is the rate at which r is changing 
it is evident that fiv is the rate at which the sidewise velocity, &r is changing, as 





Fig. 67. 



Fig. 68. 



explained in Art. 17. The relation a = 2&v is used in the discussion of motion of 
steam engine governors, where the governor balls have a motion of rotation com- 
bined with a motion towards or away from the axis. 

Consider now a circular disk A B, Fig. 69, spinning at angular velocity w about 
its axis of figure 0, and let the axis O be turning about CD at angular velocity fi. 
Consider a sample particle m of the disk at a distance r from O as shown in Fig. 
69. The velocity of m is rw, and the component of this velocity which is away 
from the axis CD is rw sin Q; and, therefore, the precessional rotation about the 
line CD involves an acceleration of m towards the reader which, according to 
equation (v), is 

a = 2wl2rsin0 (vi) 

It may be easily seen that this acceleration is towards the reader in quadrants 1 
and 2, and away from the reader in quadrants 3 and 4, and, therefore, the forces re- 
quired to produce these accelerations constitute a torque about the axis EF as 
indicated by the arrow T. 

73. Examples of precessional rotation.* (a) The precession of 
the earth's axis. The attraction of the sun keeps the earth in 

*A description of various practical aspects of gyrostatic action is given in an 
article by W. S. Franklin in the Popular Science Monthly of July, 1909. This 
article contains in particular a description and explanation of the Brennan gyro- 
static mechanism for maintaining the equilibrium of a monorail car, and the article 
also includes a description of the gyrostatic action of the boomerang. 



i ;o 



ELEMENTS OF MECHANICS. 



its orbit. The force of attraction of the sun upon the bulging 
equatorial portion a, Fig. 70, is more than sufficient to constrain 
this portion of the earth to its circular orbit around the sun, and 




Fig. 69. 

the force of attraction of the sun on the bulging equatorial por- 
tion b is less than sufficient to constrain that portion of the earth to 



g sun J-- 



plane of earth's orbit 




earth 



Fig. 70. 

its circular orbit around the sun. Therefore, the earth is acted 
upon by an unbalanced torque about which causes the earth's 
axis to describe a cone about the line V which is at right angles 



ROTATORY MOTION. 171 

to the plane of the earth's orbit. The action of the moon is 
here ignored for the sake of simplicity. 

(b) A coin rolling along the floor is, of course, rotating, and the 
instant the coin begins to be inclined to either side, the unbal- 
anced torque due to gravity causes a precessional movement 
of the axis of the coin, and the coin describes a curved path in 
consequence of this precession. 

(c) Rotating parts of machines on ship-board. The pitching 
and rolling of a vessel at sea causes, at each instant, a certain 
angular velocity 12 of the axis of a rotating machine part, and 
an unbalanced torque is immediately brought into existence. 
For example, when a steamer turns round, the propeller and 
propeller shaft change direction continuously; when a steamer 
rolls, the axis of a dynamo armature which is athwart ship 
changes its direction periodically; when a steamer pitches, the 
axis of the propeller and propeller shaft changes its direction 
periodically. In the case of a steamer driven by steam turbines 
the propeller shaft turns at high speed and the rotating member 
of the turbine is quite heavy, so that the pitching motion of 
such a vessel produces excessively large forces at the bearings 
which support the shaft. A turbine torpedo-boat of the British 
Navy went down in a heavy sea in 1899 or 1900, being probably 
broken in two by the very great forces produced by the pitching 
of the boat, and the consequent angular motion of the propeller 
shaft, forces which, perhaps, were not duly considered in the de- 
signing of the hull and supporting structure of the shaft. 

(d) The gyrostatic action of an automobile engine. Figure 71a 
represents a view of an automobile as seen from above, WW being 
the fly-wheel of the engine with its axis or shaft ab. The auto- 
mobile is represented as turning to the right as indicated by the 
curved arrow 12. During a short interval of time the angular 
velocity of spin turns from the direction OA to the direction 
OB in Fig. 71&, and the line AB represents what is to be con- 
sidered as the increment of angular velocity of spin during the 
interval, so that a torque T must act on the engine shaft as the 



172 



ELEMENTS OF MECHANICS. 



automobile turns to the right. To exert the torque T as repre- 
sented in Fig. 7 1 b, a force must push upwards on the front end a 
of the engine shaft and a force must push downwards on the 
rear end b of the engine shaft, or, in other words, the front end a 
of the engine shaft must push downwards on its bearing and the 




A_AU)Jt T 



U)+ACd 




Fig. 71a. 



Fig. 71&. 



rear end b of the engine shaft must pull upwards on its bearing. 
This reaction of the engine shaft, while the automobile is turning 
as shown in Fig. 71a, tends to push the front wheels of the auto- 
mobile against the ground with excessive force, and to lift the 
rear wheels off the ground. 

(e) The use of the gyroscope (or gyrostat as it is sometimes called) , 
for preventing the rolling of a ship at sea. A rapidly rotating 
wheel is hung from a hinge so that its axis may swing back and 
forth in a vertical plane, a vertical plane including the keel of the 
boat, as shown in Fig. 72a. The lower end of the axis is attached 
to a rod which connects to a piston in what is called a dash-pot. 
When the vessel rolls about the keel as an axis, the axis of the 
gyrostat oscillates back and forth, and the effect of the friction 
of the piston in the dash-pot is the same as if the rolling of the 



ROTATORY MOTION. 



173 



ship were hindered by excessive friction, and thereby the motion 
of rolling is greatly reduced. A small German torpedo boat, 
115 feet long by 12 feet beam, was recently equipped with a gyro- 




V////////////A 



Fig. 72a. 



stat arranged as shown in Fig. 72a.* The gyrostat wheel was 
3.3 feet in diameter, it had a mass of 1,100 pounds and it was 




Fig. 72&. 

driven at a speed of 1,600 revolutions per minute. The effect 
of this arrangement is shown in Fig. 72b, in which the ordinates 
measured from the line marked o° represent angular amplitudes 

*See paper by Otto Schlick, translated in Scientific American Supplement, for 
January 26, 1907. 



174 



ELEMENTS OF MECHANICS. 



of rolling oscillations. To the right of the point A the curve 
shows the rolling when the gyrostat is inoperative, and to the 
left of the point A the curve shows the rolling when the gyrostat 
is in action. 



KINEMATICS OF A RIGID BODY* 
74. Motion of a rigid body in a plane.— A rigid body is said to move in a plane 
when all points of the body which lie in the plane remain in it. For example, a 
rotating wheel moves in a plane, the connecting rod of a steam engine moves in a 




Fig. 73- 



Fig. 74. 



plane. Consider a rigid body A B, Fig. 73, moving in the plane of the paper. The 
position of the body is completely indicated by the position of the line A B fixed in 
the body. This line is called the index line. 

After any change in the position of a rigid body moving in a plane, a certain line 
in the body, perpendicular to the plane, is in its initial position, and the given displace- 
ment is equivalent to a rotation about that line as an axis. Let AB and A'B', 
Fig. 74, be the positions of the index line before and after the displacement. Join 
AA' and BB'. Erect perpendiculars from the middle points of AA' and BB' 
intersecting at p. From the similarity of the triangles pAB and pA'B' it is 
evident that the same part of the body is at p before and after the displacement, 
and that the line through p perpendicular to the paper is the line about which 
the body may, by simple rotation, move from its initial to its final position. The 
angle A0 of this rotation is the angle subtended by A A' or BB r as seen from p. 

75. The instantaneous motion of a rigid body moving in a plane in any manner 
is a motion of rotation about a definite line called the instantaneous axis of the 
motion. Let the displacement, shown in Fig. 74, be that which takes place in a 

*The discussion of the dynamics of a rigid body should properly be preceded by a 
discussion of the kinematics of a rigid body. This, however, has not been done 
because most of the discussion of the dynamics of a rigid body can be based upon 
the simple idea of rotation about a fixed axis. Thus the rotatory motion of a rolling 
wheel is in no way different from what it should be if the translatory motion did not 
exist. 



ROTATORY MOTION. 



175 



short interval of time At; then A<pjAt is the instantaneous angular velocity of the 
body, and the line through p, perpendicular to the paper, is the instantaneous axis. 
During a finite interval of time the motion of a body may be irregular, but the 
motion of a body during an interval of time approaches uniformity as that interval 
approaches zero. Therefore the motion of a body during a short interval of time 
is the simplest motion which can produce the actual displacement which occurs 
during the interval. 

76. Composition of angular and linear displacements. Consider an angular 
displacement A<p of a body about the point p, Fig. 75, bringing the point to 0'; 
and the linear displacement Al parallel and equal to O'O, bringing 0' back to 0. 
These two displacements are together equivalent to an angular displacement A<p 
about 0, bringing Op to Op'. Let the distance of p from the line 00' her; then, 
if A0 is small, Al = r A(f>. 

77. Resolution of motion in a plane. From Arts. 75 and 76 it follows that the 
instantaneous motion of a rigid body in a plane may be resolved into a motion of 
rotation about an arbitrary point combined with a certain linear velocity. Consider 
the actual displacement represented in Fig. 75, namely, a rotation about bringing 
p to p' '. This displacement is equivalent to an equal angular displacement Afi, 
about the arbitrary point p, together with the linear displacement O'O or pp'* 
Let this linear displacement be Ax, and let At be the interval which elapses during 
the displacement. The actual angular velocity A<p/At about the point (the 
instantaneous center) is equivalent to an angular velocity A<fi/At about the point 
p combined with a linear velocity Ax J 'At parallel to pp'. 




Fig. 75- 



Fig. 76. 



78. Motion of a rigid body with one point fixed. If a rigid body, one point 
of which is fixed, is displaced in any manner whatever, a certain line in the body will 
be in its initial position after the displacement, and the given displacement will be 
equivalent to a rotary- movement about this line as an axis. 

Proof. Consider a spherical shell of a body having its center at a fixed point. 
Let A B, Fig. 76, be an arc of a great circle on this spherical shell; the position of 
A B fixes the position of the body, and A B is called the index line. Let the move- 
ment of the body bring AB toA'B'. Connect A A' and BB' by arcs of great 



176 



ELEMENTS OF MECHANICS. 



circles. Draw great circles bisecting A A' and BB' at right angles. The point 
p at the intersection of these circles bisecting A A' and BB' has the same position 
relative to AB and A' B' , so that this point of the shell is in its initial position, and 
the line drawn from the center of the spherical shell to the point p is the axis about 
which the given movement can be produced by rotation. 

The instantaneous motion of a rigid body about a fixed point is a motion of simple 
rotation at definite angular velocity about a definite line called the instantaneous 
axis of the motion. 

79. Vector additions of angular velocities. Consider an angular velocity about 
the axis a, Fig. 77, and another angular velocity about the axis b; the two angular 




Fig. 77- 



Fig. 78. 



velocities are together equivalent to an angular velocity about the axis c, the respec- 
tive angular velocities being proportional to the lengths of the lines a, b and c. 

Outline of proof. Imagine a sphere constructed with its center at O, Fig. 77, 
and let a and b, Fig. 78, be the points where the lines a and b, Fig. 77, cut the sphere. 
Imagine a very small rotation Aa about Oa followed by a very small rotation 
A& about Ob, bringing the great circle ab to the position a'b'. The point of 

a 




Fig. 79- 

intersection of ab and a'b' is the point where the resultant axis Oc cuts the sphere, 
and the angle Ac is the amount of rotation about Oc which is equivalent to the 
two rotations Aa and A&; then it can be shown* that the three angles Aa, A& and 

*This matter is discussed in detail in Spinning. Tops by Harold Crabtree (Long- 
mans, Green, & Co., 1909) pages 34-35. 



ROTATORY MOTION. 177 

Ac are related to each other as the lengths of the three lines a, b and c, Fig. 77, and 
that the two arcs ac and cb subtend angles which are equal to <t> and 6 respec- 
tively of Fig. 77. 

80. Vector addition of torques. Let the lines a and b, Fig. 79, represent two 
given torques and let it be required to show that a and b are together equivalent to 
the torque c. Draw the lines 1-2, 2-3, and 1-3 perpendicular to and bisected by 
a, b and c respectively. The lengths of these lines are proportional to the lengths 
a, b and c. Imagine the torque a to be due to a unit upward force at 1 and a unit 
downward force at 2 (upward and downward being perpendicular to the plane of the 
paper), then the torque b is equivalent to a unit of upward force at 2 and a unit of 
downward force at 3; but the upward force and downward force at 2 annul each 
other, so that we have left only a unit of upward force at 1 and a unit of downward 
force at 3, which constitute a torque about the line c proportional to the length of c. 

Problems. 

95. A body starts from rest and after 10 seconds it is rotating 
55 revolutions per second. What is the average angular accel- 
eration? Express the result in radians per second per second. 
Ans. 34.6 radians per second per second. 

96. In what terms is moment of inertia expressed: (a) When 
length is expressed in centimeters and mass in grams? (b) When 
length is expressed in inches and mass in pounds? (c) When 
length is expressed in feet and mass in pounds? The unit 
moment of inertia in case (a) is the c. g. s. unit. How many 
c. g. s. units of moment of inertia are there in the unit involving 
the inch and the pound, and in the unit involving the foot and 
the pound? Ans. (a) grams X centimeters squared; (b) pounds 
X inches squared ; (c) pounds X feet squared. One pound-inch 3 
= 2,926 gram-centimeters 2 ; one pound-foot 2 = 421,300 gram- 
centimeters 2 . 

97. Calculate the moment of inertia of a uniform slim rod, 
length 3.1 feet (= I) and mass 3.6 pounds (= m), about an axis 
passing through the center of the rod and at right angles to the 
length of the rod. 

(a) Calculate K from the formula K = ml 2 / 12. 

(b) Calculate K approximately by multiplying the mass of 
each 0.1 foot of the rod by the square of its estimated mean dis- 
tance from the center of the rod. 

13 



178 ELEMENTS OF MECHANICS. 

(c) Calculate the radius of gyration of the rod. 
Ans. (a) K = 2.883 pounds X feet squared; (b) 2.879 pounds 
X feet squared; (c) radius of gyration 0.895 f ee t- 

98. Calculate the moment of inertia of a circular disk, radius 
1.7 feet, mass 4.25 pounds, about the axis of figure. 

(a) Calculate K from the formula given in the table in Art. 62. 
The circular disk is of course a very short cylinder. 

(b) Calculate the radius of gyration of the disk. Ans. K = 
6.14 pounds X feet squared; radius of gyration of disk is 1.22 
feet. 

99. (a) Calculate the moment of inertia of the rod, problem 
97, about an axis passing through the end of the rod and perpen- 
dicular to the rod. 

(b) Calculate the moment of inertia of the disk, problem 98, 
about an axis passing through the edge of the disk parallel to 
the axis of figure of the disk. Ans. (a) 11.52 pounds X feet 
squared, (b) 18.42 pounds X feet squared. 

100. A circular disk 5 feet diameter, weighing 1,200 pounds is 
mounted upon a shaft 6 inches in diameter. The disk, set ro- 
tating at 500 revolutions per minute and left to itself, comes to 
rest in 75 seconds. Calculate average (negative) angular accel- 
eration while stopping, calculate average torque acting to stop 
the disk, and calculate the frictional force at the circumference 
of the shaft. Ans. 0.698 radians per second per second; torque 
acting to stop the disk is 81.7 pound-feet; frictional force 327 
pounds-weight. 

101. What is the kinetic energy of the disk specified in problem 
100 when the speed is 500 revolutions per minute? Ans. 160,640 
foot-pounds. 

102. A metal disk 12 inches in diameter and weighing 25 
pounds, has a cylindrical hub projecting on each side. Each hub 
is 1 inch in diameter and weighs % of a pound (total mass 25.5 
pounds). What is the moment of inertia of the whole? 

The hubs of this disk roll on a track which drops 1 inch ver- 
tically in each foot of horizontal distance ; find how fast the disk 



ROTATORY MOTION. 1 79 

gains linear velocity in rolling down this track. Ans. 3.1254 
pounds X feet squared; 0.037 feet per second per second. 

103. A slim rod 2 feet long and having a mass of 2.5 pounds is 
suspended by a wire. The wire is attached to the middle of the 
rod and the rod hangs in a horizontal position. The rod, set 
vibrating about the wire as an axis, makes 50 complete vibrations 
in 10 minutes 25 seconds. What torque would be required to 
twist the wire through one complete turn? Ans. 0.0414 pound- 
feet. 

104. An irregular body is suspended by the same wire that is 
specified in problem 103, and, set vibrating about the wire as an 
axis, it makes 37 complete vibrations in 10 minutes. What is its 
moment of inertia? Ans. 1.404 pounds X feet squared. 

105. A uniform slim rod 4 feet long is hung as a gravity pen- 
dulum at a point distant 6 inches from the end of the bar. Cal- 
culate its equivalent length as a pendulum. Ans. 2.39 feet. 

106. A pendulum clock rated in Boston and carefully trans- 
ported to Hammerfest would gain how many seconds per day? 
Ans. 97 seconds gained per day. 

See table in Art. 70. 

107. The connecting rod of a steam engine weighs 19.5 pounds. 
Its center of mass is distant 1 8 inches from the center of the hole 
which fits the crank pin, and when it is swung as a gravity pen- 
dulum about the point a, Fig. 107 p, it makes 100 complete 



N 18 inches 

Fig. io7£. 



3$K 



■M 



vibrations in 2 minutes and 50.2 seconds. The diameter of the 
hole at a is 1 ^ inches. What is the moment of inertia of the 
connecting rod about its center of mass? Ans. 24.2 pounds X 
feet 2 . 



1 80 ELEMENTS OF MECHANICS. 

Note. The moment of inertia of the connecting rod about point a may be 
calculated from the equation of the pendulum. The moment of inertia about the 
center of mavss may then be calculated as explained in Art. 64. 

108. A slim stick 5 feet long and having a mass of 10 pounds is 
held in a vertical position and struck a horizontal blow with a 
hammer at a point 18 inches from the upper end, the upper end 
being released at the instant of the blow. The impulse of the 
blow is 80 pounds-weight-seconds. Find the translatory velocity 
imparted to the stick (velocity of its center of mass), find the 
angular velocity about its center of mass, and find the position 
of the point in the stick which remains stationary for a moment 
after the hammer blow. Ans. 257.6 feet per second; 123.6 
radians per second; the point is 2.083 feet below the middle of 
the stick. 

Note. To solve this problem it is best to reduce the impulse of 80 pounds- 
weight-seconds to poundal-seconds by multiplying by 32.2. Then the trans- 
latory velocity of the stick is found by dividing the impulse in poundal-seconds by 
the mass of the stick in pounds. Multiplying the impulse in poundal-seconds by 
the distance between the center of mass of the stick and the point where the hammer 
strikes gives the torque impulse in poundal-foot-seconds, and dividing this torque 
impulse by the moment of inertia of the stick gives the angular velocity in radians 
per second. 

109. Find the distance from the axis of suspension of the slim 
rod described in problem 105, to the point where the rod may be 
struck horizontally with a hammer without causing a side force 
to be exerted on the axis of suspension. Compare this distance 
with the "equivalent length" of the rod as a gravity pendulum. 
Ans. 2.39 feet; it is equal to the "equivalent length" of the rod 
as a gravity pendulum. 

110. A water wheel is connected to its belt-pulley by a shaft. 
Find the torque, in pound-feet and in pound-inches, tending to 
twist the shaft when the water wheel develops 200 horse-power 
at a speed of 600 revolutions per minute. Ans. 1,750 pound-feet, 
or 21,000 pound-inches. 

Note. The two equations W=T<p and P=To> correspond exactly to equa- 
tions (24) and (25) as explained in Art. 66. The only difficulty involved in the use 
of the equations W=T4> and P=Tu is to keep the units straight, as it were. 



ROTATORY MOTION. 151 

111. An electric motor, running at 900 revolutions per minute, 
develops 15 horse-power. Find the torque with which the field 
magnet acts upon the rotating armature, neglecting friction. Ex- 
press the result in pound-feet and in pound-inches. Ans. 87.6 
pound-feet, or 1,051 pound-inches. 

112. The armature shaft of a ship's dynamo is athwart ship, 
and the armature is driven clockwise as seen from the port side 
of the vessel. Describe accurately the forces with which the 
bearings act upon the armature shaft as the vessel rolls. Specify 
the directions of these forces when the port side of the vessel is 
rising, and when the the port side of the vessel is falling. 

This problem refers to the forces which arise from the rotatory 
motion of the armature. The port side of a vessel is on the left 
hand of a person facing the bow. Ans. When the port side of the 
ship is rising, the precessional torque exerted by the bearings 
upon the armature shaft is such that its reactions tend to turn 
the ship to port; the torque is reversed when the port side of 
the ship is falling. 

113. A side-wheel steamboat is suddenly turned to port, and 
the gyrostatic action of the paddle wheels causes the boat to list. 
In which direction does the boat list, to starboard or port? Why? 
Ans. To starboard. 

114. The vessel described in problem 113 is steered in a circle 
150 feet in radius at a velocity of 25 feet per second, and the 
vessel lists 5 because of the gyrostatic action of the paddle 
wheel and shaft. To produce a 5 list when the boat is stand- 
ing still requires a weight of 10 tons to be shifted from the center 
of the boat to a point 15 feet from the center. The paddle 
wheels make 75 revolutions per minute. Find the moment of 
inertia of the axle and wheels. Ans. 7,362,000 pound-feet 2 . 

115. A locomotive rounds a curve of radius 528 feet at a speed 
of 30 miles per hour. The diameter of the driving wheels is 6 
feet and each pair of drivers and the connecting axle has a moment 
of inertia of 37,000 pound-feet 2 . Find the torque acting on each 
pair of drivers due to the precession. How does this precession 



1 82 ELEMENTS OF MECHANICS. 

modify the force with which the wheels push on the two rails? 
Ans. 1,413 pound-feet; it decreases the pressure of the wheels 
on the inner rail and increases the pressure on the outer rail. 

116. A torpedo boat makes a complete turn in 84 seconds 
and its propeller rotates at a speed of 270 revolutions per minute. 
The moment of inertia of the propeller is 2,000 pound-feet. 2 
Required the precessional torque on the propeller shaft. In 
what direction does this torque tend to bend the shaft? Ans. 
132.2 pound-feet. This torque will be in such a direction as to 
depress the stern and raise the bow of the boat if the propeller 
is rotating in a clockwise direction as viewed from behind and if 
the boat is turning to port. 

117. A high speed engine with its shaft athwart ship, makes 240 
revolutions per minute. The rim of the fly-wheel has a radius 
of 3 feet and a mass of 600 pounds. Calculate the moment of 
inertia of the wheel (rim). The maximum angular velocity at- 
tained by the vessel in rolling is -j 1 ^- radian per second. Calculate 
the maximum torque acting on the fly-wheel shaft due to gyro- 
static action. Ans. 5,400 pound-feet 2 ; 424.1 pound-feet. 



CHAPTER VII. 

ELASTICITY (STATICS). 

81. Stress and strain. When external forces act upon a body 
and tend to change its shape, the body is distorted more or less, 
and the external forces are balanced by the tendency of the dis- 
torted body to return to. its original shape. The distortion of a 
body always brings forces into action between the contiguous 
parts of the body throughout. These force actions between con- 
tiguous parts of a distorted body are called internal stresses; and 
the total reaction of the distorted body, which balances the ex- 
ternal distorting force, is called the integral stress of the body. 

The actual movement of the point of application of an external 
force which distorts a body is called the integral strain of the 
body, and the change of shape of each small part of the distorted 
body is called the internal strain. Thus the elongation of a wire 
under tension, the shortening of a column under compression, 
the angular movement of the end of a rod under torsion, the de- 
pression of the middle of a beam which is loaded at its center, 
and the decrease of volume of a body which is subjected to 
hydrostatic pressure are integral strains; and the total stretch- 
ing force acting on the wire, the total load on the column, the 
total torque tending to twist the rod, the total load at the middle 
of the beam, and the hydrostatic pressure which acts on a body, 
are integral stresses. In each. of these cases, furthermore, each 
small part of the body is distorted, and force actions exist be- 
tween contiguous parts of the body throughout. These are 
called the internal strains and the internal stresses respectively. 

82. Homogeneous and non-homogeneous stresses and strains. 
It is generally the case in a distorted body, that each small part 
of the body is differently distorted, and that the internal stress 
varies from point to point in the body. For example, the dif- 

183 



1 84 



ELEMENTS OF MECHANICS. 



ferent parts of a bent beam, or of a twisted rod, are differently dis- 
torted, and the internal stress varies from point to point; the 
pressure of the atmosphere decreases and the air becomes less 
and less dense with increasing altitude above the level of the 
sea; the pressure at a point in a body of water increases with the 
depth beneath the surface, and the water is more and more com- 
pressed as the pressure increases; the stress in a long cable, 
which is suspended in a mine shaft, increases from the lower end 
upwards, and the extent to which each portion of the cable is 
stretched increases with the stress. 

When the force action between contiguous parts of a body is 
the same at every point in the body, the stress is said to be 




wire under 
tension 



column under 



body A under 
-hydrostatic pressure 



Fig. 80. 



homogeneous] and when every part of a body is similarly dis- 
torted, the strain is said to be homogeneous. Thus each part of 
a rod under tension or compression is similarly distorted as shown 
in Figs. 86a and 86b, and the force action or stress is the same at 
every point as shown in Figs. 85a and 856; the steam in a boiler 
is under the same pressure throughout (gravity negligible), and 
the degree of compression of every portion of the steam is the 
same; the water in the high pressure cylinder of a hydraulic 



ELASTICITY (STATICS.) 



I8 5 



press is under the same pressure throughout (gravity negligible), 
and the degree of compression of every portion of the water is 
the same. 




Fig. 81. 



The distinction between homogeneous and non-homogeneous 
strains is shown in Figs. 81, 82, and 83, which are photographs 



■ 











CD W^^^ c~^ 

cd oNp 

CD \ 
cd . o> \ 


( 


CD 


CD CD J 
CD J 


■ 
■ 


CD 


CD CD y 

CD CD^p^v 



Fig. 82. 



of a thin rubber sheet upon which a large circle and a number of 
small circles were drawn. Figure 81 shows the unstrained sheet 



i86 



ELEMENTS OF MECHANICS. 



Fig. 82 shows the sheet homogeneously strained, and Fig. 83 
shows the sheet non-homogeneously strained. The small circles 
are changed to ellipses in Fig. 82 and in Fig. 83, but in Fig. 82 
the ellipses are all alike and their axes are in the same direction, 
whereas, in Fig. 83, some of the ellipses are more elongated than 
others and their axes are not parallel. In a homogeneous strain 
a large portion of a substance is distorted in a manner exactly 
similar to the distortion of each small portion of the substance. 
This is shown in Fig. 82 in which the large ellipse is exactly the 
same shape as the small ones. In a non-homogeneous strain a 
large portion of a substance is irregularly distorted. This is 




Fig. 83. 

shown in Fig. 83 by the irregular curve into which the large 
circle has been converted by the strain. It is a fact of funda- 
mental importance in the theory of elasticity, that, however irregularly 
a body may be distorted, any small portion of the body suffers that 
simple kind of distortion which changes a sphere into an ellipsoid, 
or which, in the case of a thin sheet of rubber, changes a circle into 
an ellipse. That is, the change of shape of any small portion of 



ELASTICITY. (STATICS.) 1 87 

a distorted body consists of an increase or decrease of linear 
dimensions in three mutually perpendicular directions, and, in 
some cases, this simple kind of distortion is accompanied by a 
slight rotation of the small parts of the body. Thus, in Fig. 90, 
which represents a portion of a bent beam, the short straight 
lines were all horizontal or vertical in the unbent beam. 

The effect of a sharp groove in a body which is under stress is 
a matter of very great practical importance. The effect is, in 
general, to produce an excessive concentration of stress in the 
material at the bottom of the groove, and a crack or fracture is 
almost sure to develop, unless the material is plastic so that the 
bottom of the groove is broadened by yielding. Consider, for 
example, a beam in which a sharp groove is cut, as shown in 
Fig. 84. The fine lines in this figure represent the lines of stress 




Fig. 84. 

when the beam is bent, and the crowding together of these lines 
as they pass under the groove represents the concentration of 
stress above referred to. 

The most striking illustration of the effect of a sharp groove in 
a body under stress is furnished by a piece of glass in which 
there is a minute crack. A piece of glass without a crack will 
stand a very considerable stress, but if the stress "flows" round 
the end of a crack, the stress is concentrated and the crack 
extends indefinitely. A pane of window glass or a glass tumbler 
is worthless when a crack once starts. A less familiar illustra- 
tion of the effect of a sharp groove is furnished by the method 
commonly employed for breaking a bar of steel; thus a steel 
rail, which normally withstands the tremendous stresses due to 
the weight of a locomotive, can be broken in two by a hammer 



1 88 ELEMENTS OF MECHANICS. 

blow if a nick is made across the top of the rail with a sharp 
chisel. Sharp re-entrant angles are always carefully avoided in 
the designing of those parts of structures which are intended to 
sustain stress. 

83. Solids and fluids. Everyone is familiar, in a general way, 
with the three classes of substances, solids, liquids, and gases. 
A solid can withstand, for an indefinite length of time, a stress 
which tends to change its shape. A solid which recovers from 
distortion (strain) when stress ceases to act, is said to be elastic. 
Thus good spring steel recovers almost completely from a moder- 
ate amount of distortion when the distorting force (stress) ceases 
to act. A solid which does not recover from strain when the 
stress ceases to act, is said to be plastic. Thus lead and wax are 
plastic solids. No solid, perhaps, recovers completely from dis- 
tortion; and, on the other hand, every plastic substance, perhaps, 
is slightly elastic. Thus the best spring steel does not completely 
recover from even a slight distortion, and when the distortion is 
great the steel takes a very decided permanent set ; and even wax 
is slightly elastic, as is shown by the distinct metallic ring of 
a large cake of beeswax or paraffine when it is struck with a 
hammer. 

A fluid is a substance which, at rest, cannot sustain a stress 
which tends to change its shape. While a fluid is actually chang- 
ing shape, however, it does sustain a stress which tends to change 
its shape. Thus a stream of syrup falling from a vessel is under 
tension like a stretched rope, and the effect of this tension is to 
continually lengthen each portion of the stream of syrup as it 
falls. A fluid at rest always pushes normally against every por- 
tion of a surface which is exposed to the action of the fluid. 
Thus the steam in a boiler pushes outwards on the boiler shell 
at each point, the water in a vessel pushes normally against the 
walls of the vessel at each point, and the atmosphere pushes 
normally against every portion of an exposed surface. A fluid 
in motion, however, may not push normally against an exposed 
surface. Thus, a water pipe is subject only to a bursting force 



ELASTICITY. (STATICS.) 1 89 

if the water is at rest; but if the water flows through the pipe, 
it has a slight tendency to drag the pipe along with it.* 

A liquid is a fluid, like water or oil, which can have a free sur- 
face, such as the surface of water in a glass. A gas, on the 
other hand, is a fluid which completely fills any containing vessel. 

84. Hooke's law. Elastic limit. Robert Hooke discovered, 
in 1676, that what we have called the integral strain of a body is 
quite accurately proportional to what we have called the integral 
stress, f Thus, the elongation of a wire under tension is propor- 
tional to the stretching force, the shortening of a loaded column 
is proportional to the load, the angular movement of the end of 
a rod under torsion is proportional to the torque which acts on 
the rod, and the depression of a loaded beam is proportional to 
the load. 

Elastic limit. Hooke's law is quite accurately true for dis- 
tinctly elastic substances like steel, but it does not apply to plas- 
tic substances, and even for elastic substances like steel there is 
a limit, called the elastic limit, beyond which stress and strain are 
no longer even approximately proportional. When an elastic 
substance is strained beyond its elastic limit it does not return to 
its original size or shape when the stress ceases to act, but takes 
what is called a permanent set. Liquids and gases, however, re- 
turn to their exact initial volume when relieved from pressure, 
provided the temperature has not changed, that is, liquids and 
gases may be said to be perfectly elastic, but when a liquid or 
gas is compressed the diminution of volume (integral strain) is 
not proportional to the increase of pressure (integral stress) ex- 
cept when the increase of pressure is fairly small. This is at 
once evident in the case of gases when we consider that they 
conform to Boyle's law as explained in Art. 101. 

*A jet of water issuing from the end of a pipe pushes backwards on the pipe, as 
every fireman knows. This backward force is due to the normal force with which 
the water pushes on ihe inner walls of the pipe where the pipe bends. 

tit follows from this experimental fact that the strain at each point of a dis- 
torted elastic body is proportional to the stress at that point. 



I90 ELEMENTS OF MECHANICS. 

85. Limitations and plan of this chapter. The phenomena 
which are associated with the distortion of bodies are excessively 
complicated. Let one consider the swaying of objects in the 
wind, the bending and compression of structures under load and 
their vibration with sudden variations of load ; let one think of 
all the familiar properties of brittle substances like chalk and 
glass, of plastic substances like clay and wax and of elastic sub- 
stances like steel and rubber; let one consider that all of the 
phenomena of sound are due to the vibrations of bodies, and to 
wave movements in the air, and, in many cases, to wave move- 
ments in water and in solids, all of which have to do with distor- 
tion and compression; and let one think that compression and 
expansion and local changes of shape are involved inextricably 
in nearly every case of flow of air and water. Let one think of 
all of these things and then consider whether it is not necessary 
to bring the mind to a narrow view before any clear line of 
argument can be pursued relative thereto ! 

Of all the great variety of solid substances, having almost 
every imaginable degree of elasticity, plasticity, hardness, and 
brittleness, and ranging in strength from sun dried clay to the 
toughest steel, we are chiefly concerned with the behavior under 
stress of those which are used as materials of construction; 
and, in addition, we are here concerned with the tendency of 
increase of pressure to reduce the volumes of liquids and gases. 

A substance, like wood, which has a grained structure, is said 
to be aeolotropic (pronounced e'-o-lo-trop'-ic). Most crystalline 
substances, and rolled and drawn metal are aeolotropic. A sub- 
stance, like glass or water, which does not have a grained struc- 
ture, is said to be isotropic. The behavior under stress of aeolo- 
tropic substances is very complicated; these complications need 
not be considered, however, for practical purposes, because sub- 
stances having a grained or fibrous structure are generally sub- 
jected to stresses parallel to the grain, as in beams, and ropes 
and wires. The difference between aeolotropic substances and 
isotropic substance is ignored in this chapter 



ELASTICITY. (STATICS.) 191 

Types of stress and strain. In the discussion of the behavior 
of bodies under stress, it is necessary to consider three simple 
types of stress and strain. Thus we have longitudinal stress and 
longitudinal strain, which is the type of stress and strain in a rod 
under tension or in a column under compression ; we have hydro- 
static pressure and isotropic* strain, which is the type of stress 
and strain in a body subjected to hydrostatic pressure; and we 
have shearing stress and shearing strain, which is the type of 
stress and strain which exists (non-homogeneously) in a twisted 
rod. A discussion of the first two types of stress and strain is 
sufficient for most practical purposes, and, therefore, the discus- 
sion of shearing stress and shearing strain is given in small type 
preceding the outline of the general theory of stress and strain. 

Treatment of non-homogeneous stresses and strains. In the 
following discussion, the behavior of a substance under each type 
of homogeneous stress and strain is first considered, and the ideas 
so developed are used as a basis for the discussion of important 
cases of non-homogeneous stresses and strains. For example, 
the discussion of the bent beam is based upon the discussion of 
homogeneous longitudinal stress and strain, and the discussion 
of the twisted rod is based upon the discussion of homogeneous 
shearing stress and strain. 

LONGITUDINAL STRESS AND STRAIN. 

86. Longitudinal stress. Figure 85a represents a portion of 
a rod under tension. Let F be the total force tending to stretch 
the rod, and let A be the sectional area of the rod; then the 
stretching force per unit of sectional area is F/A(=P), and the 
force action between contiguous portions of the rod is as follows: 
Imagine a horizontal unit of area q anywhere in the material of 
the rod, the material on the two sides of q exerts a pull P across 
q, as shown in Fig. 85a; imagine a vertical unit of area q r drawn 
anywhere in the material of the rod, the material on the two sides 

*When an isotropic substance, such as glass, is subjected to a hydrostatic pressure 
the substance is reduced in volume without being changed in shape. Such a strain 
is called an isotropic strain, for want of a better name. 



192 



ELEMENTS OF MECHANICS. 



of q' does not exert any force at all across q r . The force acting 
across any horizontal area of a units is, of course, equal to Pa. 
The force per unit area, P, is the measure* of the longitudinal 
stress, and the direction of P is called the axis of the stress. 

A rod under tension may be considered as a case of positive 
longitudinal stress, and a rod or column under compression may 
be considered as a case of negative longitudinal stress. 




rod under tension 

Fig. 85a. 



p 

> f 


1 

1 




1 

V 


f. 


-~9 



rod under compression 

Fig. 85&. 



87. Longitudinal strain. A rod under tension is longer than 
when it is not under tension, and, since each unit portion of the 
rod must be equally stretched, it is evident that the increase of 
length of the rod is proportional to its total initial length. There- 
fore it is most convenient to express the increase of length as a 
fraction of the total initial length; thus, a stretch of 2 thou- 
sandths means an increase of length equal to 2 thousandths of 
the total initial length of a rod. Let L be the initial length of 
a rod and let / be its increase of length under tension, then 
1 1 L(= /3) is the increase of length expressed as a fraction of the 
initial length, and, of course, the ratio of L + / to L is equal to 
1+ |8. The fraction /3(= 1/ L) is used as the measure of the 
longitudinal strain. 



*That is, the number which is used to specify the value of the stress. See Art. 9. 



ELASTICITY. (STATICS.) 



193 



The character of the distortion of the parts of a rod under ten- 
sion is shown in Fig. 86a, any spherical portion of the material 
of the rod becomes an ellipsoid of revolution. The major axis 
of the ellipsoid is 1 + /3 times . the diameter of the original 
sphere. 

Lateral contraction of a stretched rod. Pois son's ratio. When 
a rod is stretched, it contracts laterally as indicated by the dotted 




rod under tension 

Fig. 86a. 




rod under compression 

Fig. 86&. 



ellipses in Fig. 86a. This lateral contraction of a stretched body 
is very strikingly shown by a stretched rubber band. The lateral 
contraction d of a stretched rod is most conveniently expressed 
as a fraction of the original diameter D of the rod, and the frac- 
tion d/D we may represent by the letter /3'. The elongation 
of a rod per unit length (/3) bears, for a given substance, a fixed 
ratio to the lateral contraction per unit of diameter (jS'), and this 
ratio is called Poisson's ratio; (3 is approximately four times as 
large as &' for steel, brass, and copper. 

88. Stretch modulus of a substance. The elongation of a 

rod under tension is proportional to the stretching force (Hooke's 

law). Therefore the stretching force divided by the elongation 

is a constant for a given rod, and if we divide stretching force 

14 



194 ELEMENTS OF MECHANICS. 

per unit area, P, by elongation per unit of original length /5( = Z/L), 
the result is a constant which depends upon the material of the 
rod, but which is independent of the size and length of the rod. 
This constant, which is represented by the letter E, is called 
the stretch modulus* of the material of which the rod is made. 
That is 

E = | (48) 

The stretch modulus may be defined in a slightly different way 
as the factor which, multiplied by the elongation per unit length, 
gives the stretching force per unit sectional area of a rod under 
tension (Efi = P) ; and it is evident from this definition that E 
is expressed in units of force per unit of area, inasmuch as /3 is a 
ratio of two lengths (l/L); in fact E is the force per unit sec- 
tional area which would double the length of a rod if the elonga- 
tion would continue to be proportional to the stretching force. 
This, of course, is not true for elongations of more than a few 
parts per thousand. 

89. Determination of stretch modulus. The stretch modulus 
may be determined by applying a known stretching force F to a 
rod of known length L and known sectional area A , and observ- 
ing the increase of length I. Then P = F/A, and (3 = 1/ L, 
so that E = P/p = FL/Al. An easier method for determining 



TABLE. 


Values of the stretch modulus of various substances. 


(In pounds- weight per square inch.) 


Copper (drawn) 


17,700,000 


Steel (rolled) 


29,800,000 


Wrought iron 


29,600,000 


Cast iron 


16,000,000 


Glass 


9,600,000 


Oak wood 


1,450,000 


Poplar wood 


750,000 



the stretch modulus of a substance is by observing the deflection 
of a loaded beam as explained in Art. 91. 

*Often called Young's modulus, or "the modulus of elasticity" by engineers. 



ELASTICITY. (STATICS.) 



195 



90. Potential energy of longitudinal strain. The potential 
energy per unit volume of a stretched (or compressed) rod is 
equal to one half the product of the stress P and the strain /3, or 
it is equal to one half of the product of the stretch modulus E 
and the square of the strain /?. That is 



/ 2 P(3 



or 



W 

W = y 2 E(3 2 



(49) 
(50) 



in which W is the potential energy per unit of volume of a 
stretched (or compressed) rod, P is the stretching (or com- 
pressing) force per unit sectional area of the rod; /3 is the in- 
crease (or decrease) of length per unit of original length, and E 
is the stretch modulus of the material. If P and E are expressed 
in pounds- weight per square inch, W is expressed in inch-pounds 
of energy per cubic inch of material. 



Axis otF 




Proof. The increase of length I of a rod is proportional to the 
stretching force F, so that by plotting corresponding values of / 
and Fwe have a straight line OA, as shown in Fig. 87. 

Imagine the stretching force to increase slowly from zero to F, 
the stretch at the same time increasing from zero to I. Let F f 
be an intermediate value of F, and let Al be a very small increase 
of I due to a slight increase of F f , then F f -Al is the work done 
on the rod during the very slight increment of stretch Al. But 
F'-Al is the area of the narrow parallelogram shown in Fig. 87, 



196 



ELEMENTS OF MECHANICS. 



and therefore the total work done on the rod while the stretching 
force increases from zero to F is the total area OAB Fig. 87, 
which is equal to yi Fl, so that the work done per unit volume of 
the rod is }iFl divided by the volume AL of the rod. That is 

, IF 



W 



AL 



4P(3 



This proof may be stated in a slightly different way thus: The 
average value of the stretching force between zero stretch and the 
given stretch I, is %F, which, multiplied by the elongation /, 
gives the work done on the rod. 

Equation (50) is derived from equation (49) by substituting 
Efi for P according to equation (48). 

91. Discussion of a bent beam.* The simplest case of a bent 
beam is that which is shown in Fig. 88 in which a long beam is 
laid horizontally across two supports SS, and bent by hanging 
weights on the projecting ends as shown. The bending moment 
or torque of each weight is equal to Wx, and this bending torque 
acts on every portion of the beam between the supports SS, so 
that this portion of the beam becomes an arc of a circle, f All of 

*When a beam is bent the stretched filaments on one side of the beam contract 
laterally and the compressed filaments on the other side of the beam expand laterally 
and the section of a beam originally square becomes distorted somewhat as shown in 
Fig. 89. This is very clearly shown by bending a rectangular bar of rubber, a lead 





Section of beam before bending. Section of beam after bending (exaggerated) . 

Fig. 89. 
pencil eraser for example. This distortion of the section of a bent beam is usually 
very slight and it is neglected in the above discussion. 

fTo show that the bending moment to which the beam is subjected has the 
same value everywhere between the supports 55 in Fig. 88, consider the portion 
of the beam aaa/a' . This portion of the beam is in equilibrium. The downward 
force W and the equal upward force at the support S constitutes a pure torque 
of which the value is Wx, and therefore the force with which the portion is acted 
upon at a!a f by the remainder of the beam, that is the force action across the 
section a / a / , is a pure torque which is equal and opposite to Wx. 



ELASTICITY. (STATICS.) 



I 9 7 



the filaments in the upper part of the beam are elongated, all of 
the filaments in the lower part of the beam are shortened, and 
certain filaments pp, which lie in what is called the median line 
or surface of the beam, remain unchanged in length. The beam 




is everywhere under longitudinal strain as shown in Fig. 90 ; and 
the force action between contiguous parts of the beam is as shown 
in Fig. 91. These figures may be understood by comparing them 




Fig. 90. 

with Figs. 85 and 86. To find the value of the stress and strain 
at each point in the bent beam proceed as follows : 

Consider the portion of the beam which lies between the radii 
R and R, Fig. 88. This portion is shown to a larger scale in 
Fig. 92. Let R be the radius of curvature of the median line pp, 
then the length of pp is equal to Rd which is the original length of 



198 



ELEMENTS OF MECHANICS. 



every filament of the beam between the radii R and R, Fig. 88. Con- 
sider a filament of the beam at a distance y above the median line, 




Fig. 91. 

y being considered negative for filaments below the median line. 
The radius of curvature of this filament is R + y and its length is 
(R + y)9. Therefore, the increase of length of this filament due 
to the banding of the beam is yd, and, expressing this increase 
of length as a fraction of the original length Rd, we have 



y 



(i) 



which expresses the value of the longitudinal strain at any point 




Fig. 92. 



in the bent beam, /3 being positive where y is positive and negative 
where y is negative. 



ELASTICITY. (STATICS.) 



199 



The longitudinal stress P (force per unit of area, as shown in 
Fig. 91) at each point of the beam is equal to Eft, according to 
equation (48), where E is the stretch modulus of the material of 
the beam. Therefore, using y/R for (3, we have 



P 



E 
R 



•y 



(ii) 



The total force action across a complete section ab of the 
beam shown in Fig. 93, that is, the total force action of the por- 
tion A A upon the portion BB, is a torque about an axis 




perpendicular to the plane of the figure. This axis is shown as 
the line 00 in Fig. 94, which is a sectional view of the beam, 



o — 




b being the breadth of the beam and d its depth. The torque 
action is expressed by the equation 



200 ELEMENTS OF MECHANICS. 

i bd 3 E 
T = iV -^ (iii) 

But the torque action across every section of the beam between 
the two supports 55 in Fig. 88 is equal to Wx, so that 

i bd 3 E 
^ = iV-£- (iv) 

from which E may be calculated when W, x, b, d and R are 
known. 

Derivation of equation (iii). The portion of the beam between the two parallel 
lines Ay, in Fig. 94, has a sectional area equal to b • Ay, and the force action 
across each unit of area of this portion of the beam is Ey/ R, according to equation 
(ii) above. Therefore the total force action across the area b • Ay is Eby • Ay/ R, 
which, multiplied by the lever arm y, gives the torque action about 00 which is 
due to the portion of the beam b • Ay. Therefore, 

Fh 

whence by integrating between the limits y=—d/2 and y = -\-dJ2, we have 
equation (iii). 

92. Important practical relations between longitudinal stress 
and strain.* The important practical aspects of the relation be- 
tween longitudinal stress and strain may best be brought out by 
considering the behavior of a steel rod (a test piece) which is 
subjected to a continually increasing longitudinal stress. Thus 
the ordinates of the curve A in Fig. 95 represent the values of an 
increasing longitudinal stress in pounds- weight per square inch, 
and the abscissas represent the corresponding elongations, in 
hundredths, of a rod of ordinary bridge steel. The curve B is 
the left-hand part of Curve A with abscissas magnified one hun- 
dred times. 

Up to a point p, the position of which is not very sharply de- 
fined, the strain (elongation per unit initial length) is very exactly 
proportional to the stress (stretching force per unit of sectional 
area) ; that is the stress-strain curve is a straight line from the 
origin to p. Beyond the point p the stress-strain line is very 
slightly curved until the point q is reached where the steel begins 

*The student is referred to the splendid treatise on "The Materials of Construc- 
tion" by J. B. Johnson, Wiley and Sons, 1898, for a full discussion of this subject. 



ELASTICITY. (STATICS.) 



20I 



to yield very greatly. This yielding takes place rather irregu- 
larly until the whole test-piece has yielded, it alters the temper 
of the steel, and the steel then sustains an increased stress which 
reaches a maximum at the point t. The metal then begins to be 
weakened by the continued increase of length, and finally the rod 
breaks at the point b. 

The point p marks the true elastic limit; the point q, which is 
sometimes called the yield point, marks what is for practical 



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5, 10 15 20 25 3a 

Stretch, hundredths for curve A, ten-thousandths for curve B> 

Fig. 95- 

purposes* called the elastic limit, and the point t marks what is 
called the tensile strength of the steel. The most important of 
these points is the yield point or elastic limit, commerically so 
called, because the steel breaks after a very few applications of 
a stess that exceeds the elastic limit. Everyone is familiar with 
this fact in that a wire may be easily broken by bending it repeat- 
edly beyond the elastic limit, and one can easily imagine how 
short-lived a bridge or a steel rail would be if it were strained 

♦Because of the difficulty of locating the point p accurately. 



202 



ELEMENTS OF MECHANICS. 



beyond the elastic limit by every passing train. The total elonga- 
tion of a sample of steel at the breaking point b, Fig. 95, is an 
important indication of the toughness of the steel. The follow- 
ing table gives the important properties of several grades of steel. 



TABLE. 

Physical properties of steel. 



Carbon content 
in per cent. 


Elastic limit pounds 
per square inch. 


Tensile strength 

pounds per square 

inch. 


Elongation in 4 

inches in per 

cent. 


Stretch modulus in 

pounds per square 

inch. 


O.17 

0-55 
0.82 


51,000 

57,000 

63,000 


68,000 
106,100 
142,250 


33-5 
16.2 

8.S 


29,800,000 



93. Resilience. The work done per unit volume in straining 
a substance to its elastic limit is called the resilience of the sub- 
stance. This work per unit volume is equal to one half the 
product of the limiting stress and the limiting strain, according 
to equation (49), and it is represented by the area under the 
straight portion of the curve in Fig. 95. For example, the 
resilience of 0.82-per-cent. carbon steel is 

1 pounds _ inch-pounds foot-pounds 

1x63,000^^ x o.oo 2I =66.1 .^ - =S-5 inch a 

that is, 5.5 foot-pounds per cubic inch. Thus it would take 100 
cubic inches of this steel (about 25 pounds) made into a spring 
to store sufficient energy to supply one horse-power for one 
second, provided the spring could be so designed as to be strained 
in every part to its elastic limit when wound up. The resilience 
of spring steel may be as high as 10 or 12 foot-pounds per cubic 
inch, the resilience of good cast iron is about 0.5 foot-pound 
per cubic inch. 

The resilience of a substance is a measure of its strength to 
withstand a sudden shock, inasmuch as a blow of a hammer, for 
example, bends a bar until the kinetic energy of the hammer is 
all used in bending the bar. A structure subject to shocks should 
be made of highly resilient material. 



ELASTICITY. (STATICS.) 



203 



94. Elastic hysteresis. In nearly all substances there is more 
or less of a tendency for strain to persist after the stress has 
ceased. This is of course very markedly the case where a sub- 
stance is strained beyond the elastic limh , but in many substances 
the elastic limit is by no means sharply defined, and very slight 
strains do not entirely disappear when the stress ceases. When 
a substance is subjected to a stress which increases and de- 
creases periodically between two limiting values S x and S 2 , the 
relation between stress 

and strain is somewhat 
as indicated in Fig. 96, 
where ordinates repre- 
sent stress and abscissas 
represent strain. The 
branch a of the curve 
represents the relation 
between stress and strain 
while stress is increas- 
ing, and the branch b 
represents the relation 
between stress and 
strain while stress is 
decreasing. This divergence of the curve of stress and strain for 
increasing and decreasing stress is called elastic hysteresis. The 
increasing and decreasing stress is here supposed to increase and 
decrease very slowly. If the increase and decrease is rapid the 
divergence of the two curves a and b, Fig. 96, is due to hysteresis 
and also to elastic lag. 

95. Elastic lag; viscosity. Many substances, glass for ex- 
ample, when subjected to stress, take on a certain amount of 
strain quickly, after which the strain increases slowly for a time; 
and when the stress is relieved, a remnant of the strain persists 
for a time. This phenomenon is called elastic lag. 

The strain of some substances, such as pitch, continues to in- 
crease indefinitely, although it may be very slowly, when they 




Fig. 96. 



204 



ELEMENTS OF MECHANICS. 



are under stress. Such substances are said to be viscous. Nearly 
all metals are viscous when subjected to great stress. 

Elastic hysteresis, elastic lag, and viscosity cause energy to be 
dissipated in a substance when it is strained. Thus the vibrations 
of a steel spring die away rapidly even in a vacuum, on account 
of the conversion of energy into heat as the spring is repeatedly 
distorted. 

96. Elastic fatigue. The repeated application of a stress 
weakens a metal so that it will break under less than its normal 
breaking stress, or less even than the stress corresponding to its 



°v 70 

'S'S 



Ul 50 



•5* 



12 3 4 567 8 



"40 



repetitions of stress in millions 
Fig. 97- 

elastic limit. Fig. 97 shows the decrease in tensile strength of a 
sample of mild steel with repetitions of the stress. 

Continued repetition of stress causes an increase in the amount 
of energy dissipated by elastic lag and viscosity. Thus the vibra- 
tions of a torsion pendulum die away faster after it has been kept 
vibrating for several days, than at first. 

HYDROSTATIC PRESSURE AND ISOTROPIC STRAIN. 

97. Hydrostatic pressure.* A fluid at rest not only pushes 
normally against a surface which is exposed to its action, but two 
contiguous portions of a fluid at rest always push on each other 
at right angles to a small plane q which may be imagined to 
separate them as indicated in Fig. 98. Whatever the direction 
of the small plane q may be, the force action per unit area across 

*Hydrostatic pressure is discussed also in the chapter on hydrostatics. 



ELASTICITY. (STATICS.) 



205 



it is the same. This fact was first pointed out by Pascal (1623- 
1662) and it is sometimes called Pascal's principle.* The force 
action per unit area at a point in a fluid is generally represented by 
the letter p and it is called the hydrostatic pressure at the point. 

98. Isotropic strain. When a substance like glass or cast 
metal is subjected to an increase of hydrostatic pressure the sub- 
stance is reduced in size without 

being changed in shape; such a 
strain is called an isotropic strain. 
Let Vbe the original volume of the 
substance, and let v be the diminu- 
tion of volume due to the increase 
of pressure. It is convenient to 
express v as a fraction of V, and 
this fraction v/V is used as a meas- 
ure of the isotropic strain. 

99. Bulk modulus of a substance. 

The diminution of volume of 

a substance per unit of original volume (v/V) is proportional 
to the increase of hydrostatic pressure, except, of course, when 
the increase of pressure is very great and the decrease of volume 
considerable. Therefore, for small changes of volume, the ratio 
of increase of hydrostatic pressure to decrease of volume per 
unit volume is a constant for a given substance (at a given tem- 
perature). That is 

pV 
v 
V 




Fig. 



B =- = 

v 



($i)a 



in which V is the volume of a substance at a given pressure, v is 
the decrease of volume due to an increase of pressure p, and B 
is a constant for the given substance. This quantity B is called 
the bulk modulus of the substance. The reciprocal of B is called 

*This principle may be established by considering the equilibrium of a small 
portion or element of a fluid bounded by the three reference planes XY, YZ and 
ZX and by a diagonal plane. 



206 ELEMENTS OF MECHANICS. 

the coefficient of compressibility of the substance. Therefore, 
writing C for i/B equation (51 )a becomes 

c = ~ (Si)* 

or 

v = pVC (5i)c 

The coefficient of compressibility of a substance is the change 
of volume per unit original volume per unit increase of pressure, 
and, according to equation (si)c, the decrease of volume of a 
substance due to a given increase of pressure is equal to the 
product of the increase of pressure, the original volume, and 
the coefficient of compressibility. 

TABLE* 

Coefficients of compressibility at 20 C. for moderate increase of pressure. 
(Decrease of volume per unit volume per atmosphere increase of pressure.) 



Substance. 


C X 106. 


Ether 


170.0 


Alcohol 


101.0 


Water 


46.0 


Glass 


2.2 


Steel 


0.68 



100. Potential energy of isotropic strain. The potential energy 
per unit volume of an elastic substance under increased pressure 
is equal to one half the product of the increase of hydrostatic 
pressure p and the strain v/V, or it is equal to one half the 
product of the bulk modulus B and the square of the strain 
(v 2 /V~). This relation may be derived in a manner very similar 
to the proof of equations (49) and (50). 

101. Compressibility of gases. Boyle's law. Solids and 
liquids generally decrease but slightly in volume when subjected 
to increase of pressure. Thus the volume of water decreases 
about one ten-thousandth part when subjected to an increase of 

*See Physikalish- Chemische Tabellen by Landolt and Bornstein, Berlin, 1895 
for a very complete collection of data of all kinds. 



ELASTICITY. (STATICS.) 207 

pressure of 30 pounds per square inch, and the volume of steel 
decreases about one ten-thousandth part when subjected to an 
increase of pressure of 2,000 pounds per square inch. 

Gases, on the other hand, decrease greatly in volume when 
subjected to increase of pressure. The remarkable contrast be- 
tween water and air in regard to compressibility may be shown 
by filling a bicycle pump with air and then with water, and strik- 
ing the piston rod in each case with a hammer. The air will be 
found to act as a cushion, and the water will appear to be as solid 
as if the whole pump barrel and piston were one piece of steel. 
When a steam engine is started, the water which usually collects 
in the steam pipes may enter the cylinder in sufficient quantity to 
cause the moving piston to burst the cylinder head. 

When the temperature of a gas is kept at a constant value, the 
volume of the gas is inversely proportional to the pressure to which 
the gas is subjected. That is 

k 

P 

or 

pv = k (52) 

in which v is the volume of a given amount of gas, p is the pres- 
sure of the gas, and k is a constant. This relation, which is 
known as Boyle's law, was discovered by Robert Boyle * in 
1660, and more completely established by Mariotte, who discov- 
ered it independently in 1676. It is very accurately true of such 
gases as hydrogen, nitrogen and oxygen at ordinary temperatures 
and pressures, but all gases deviate from it appreciably, especially 
at low temperatures and under great pressures. See the discus- 
sion of the properties of gases in the chapters on heat. 

SHEARING STRESS AND SHEARING STRAIN. 

102. Shearing stress. The type of stress and strain in a twisted rod is called 
shearing stress and shearing strain, and the discussion of this type of stress and strain 
is somewhat obscured by the fact that there is no familiar example in which homo- 
geneous shearing stress and shearing strain occur; the stress and strain in a twisted 

*New Experiments touching the Spring of Air, Oxford, 1660. 



208 



ELEMENTS OF MECHANICS. 



rod are non-homogeneous. Any intelligible discussion of shearing stress and shear- 
ing strain must, however, be based on a case in which the stress and strain are homo- 
geneous. Consider, therefore, a cubical portion of a substance A BCD, Fig. 99, 
and suppose that outward forces (S units of force per unit of area) act upon the faces 
A B and CD, that inward forces (S units of force per unit of area) act upon the 
faces A C and BD, and that no force at all acts on the two faces of the cube which 
are parallel to the plane of the paper. Then the material of the cube will be subject 
to what is called a shearing stress, and the stress will be homogeneous. The charac- 
ter of the force action between contiguous parts of the material of the cube" is as 




4 T Y v f t 



Fig. 99. 



follows: A pull of 5 units of force per unit area acts across any plane Qx which is 
parallel to the faces AB and CD of the cube; a push of 5 units of force per unit 
area acts across any plane q 2 which is parallel to the faces AC and BD of the cube; 
a sliding force, or tangential force, as it is called, of 5 units of force per unit of area 
acts across any plane q 3 or g 4 which is parallel to the diagonal plane AD or BC; 
and no force at all acts across a plane which is parallel to the plane of the paper. 
To show that the force action across the diagonal planes q 3 and g 4 is a tangential 
force action and that the tangential or sliding force is 5 units of force per unit area, 
consider any unit cube abed of the material. The area of each face of this cube is 
unity, and the area of the diagonal plane be is V2 units. The total force acting on 
the face bd is a push of 5 units, the total force acting on the face cd is a pull of 5 
units, and the resultant of these two forces is a force parallel to be and equal to 
\2 • 5 as shown in Fig. 100. Similarly, the resultant of the forces acting on the 
faces ab and ac is a force parallel to cb and equal to V2 • 5. Therefore, the force 
action across the diagonal plane be is a tangential force action equal to \2 ■ S, 
which, divided by the area of the diagonal plane be, gives a tangential force action 
of 5 units of force per unit of area. 



ELASTICITY. (STATICS.) 



209 



It can be shown in the same way that the force action across g 4 , Fig. 99, is a 
tangential force of 5 units per unit of area. It is on account of the purely tangential 
forces across q 3 and g 4 that this type of stress is called a shearing stress. 




Fig. 100. 

103. Shearing strain. The effect of the prescribed stress in Fig. 99 is to shorten 
the cube in the direction of the push and to lengthen the cube by an equal amount 





1— A ■' ! 




1 / \ 

l\ / 

!. \ / 

i \ / ! 









1 1 
Fig. 101. 



'5 



2IO 



ELEMENTS OF MECHANICS. 



in the direction of the pull, without changing the dimensions of the cube in a direc- 
tion at right angles to the plane of the paper in Fig. 99. This distortion is rep- 
resented by the dotted rectangle in Fig. 101, and the dotted rhombus mnop in 
Fig. 101 is a figure which was square in the unstrained material. The angle 6 is less 
than 90 and the value of the angle 90° — $ (expressed as a fraction of a radian) is 
called the angle of the shearing strain, or simply the angle of shear. It is usually 
represented by the letter 0, and it is used as a measure of the shearing strain. 

Let L be the original length of each edge of the cube in Fig. 99, and let I be the 
increase of length in the direction of the pull and the decrease of length in the direc- 
tion of the push. It is convenient to express I as a fraction of L, and we will repre- 
sent this fraction by the letter a( = lf L). It is important to know that the angle 
of shear as above defined is equal to 2a. That is 



= 2tt 



(53) 



The full-line square in Fig. 102 is the figure which when distorted becomes the 
rhombus in Fig. 101, and the small triangle in Fig. 102 is enlarged in Fig. 103. The 




Fig. 102. 



Fig. 103. 



angle 0/2 is very small and it is therefore sensibly equal to gh divided by gn. 
That is, 0/2 is equal to l/L( = a) so that = 2a. 

104. Slide modulus of a substance. The angle of shear <f> produced in Fig. 99 
by the shearing stress S is proportional to S, according to Hooke's law, so that the 
ratio Sl<t> is a constant for a given substance, within the limits of elasticity. This 
constant is called the slide modulus of the substance and it is represented by the 
letter n. Therefore we have 

5 



<P 



(54) 



The slide modulus of a substance is sometimes called the shearing modulus of the 



ELASTICITY. (STATICS.) 



211 



substance. It is approximately equal to f of the stretch modulus (Young's modulus) 
for metals. 

105. Discussion of a twisted rod. Consider a cylindrical rod of radius R and 
length L, and suppose that one end of the rod is fixed while the other is turned 
through the angle 6 so as to twist the rod. Consider a cylindrical shell of the ma- 
terial of the rod of which the radius is r, and imagine this cylindrical shell to be cut 
along one side and laid out flat so that it may be pictured on a flat surface.* Figures 



<__^ 


;. 


-tnrt—.— 


*, 






\A 




"f " 














i 










JL 










B 




_l 




Fig. 104. 



Fig. 105. 



104 and 105 represent the cylindrical shell laid out flat, or developed, in this way; 
Fig. 104 before twisting, and Fig. 105 after twisting. The line A B which is parallel 
to the axis of the rod in the untwisted rod takes the position A ' B f after the rod is 
twisted, and the small square becomes a rhombus. The portions of the rod in the 
cylindrical shell under consideration are subjected to a shearing strain of which the 
angle of shear is 

rd 
<P = ~ T (55) 



This discussion of the strain in a twisted rod may be made more easily intelligible 
by means of a model as follows: A tin cylinder about 20 cm. in diameter and 35 cm. 
long, has wooden disks fixed in each end. On one end is an additional wooden disk 
which turns on a nail which is the axis of the cylinder. Loosely woven muslin is 
tacked at one end to the movable disk and at the other end to the fixed disk. This 
muslin fits the tin cylinder closely, and the seam at one side is sewed. On this 
muslin a small square may be drawn like Fig. 104, and also small circles, and when 
the movable disk is turned through a considerable angle, the distortion of the square 
and of the circles will give a clear idea of the character of the strain in a cylindrical 
shell of a twisted rod. 



212 ELEMENTS OF MECHANICS. 

as is evident from Fig. 105. The character of the force action between contiguous 
portions of the material of the rod may be understood by comparing Fig. 105 with 
Fig. 99. There is a tangential force action across every vertical plane q 3 and across 
every horizontal plane g 4 * there is a normal pull across every plane like q lt and a 
normal push across every plane like q 2 ; and the force per unit area, S, in each case is 
equal to n<j>, according to equation (54). Therefore, substituting for its value 
from equation (55), we have 

nr<t> 

The tangential stress across vertical planes like g 3 , Fig. 105, is concentrated at the 
bottom of a sharp groove cut in the rod parallel to its axis, like a key seat in a 
shaft, and such a groove therefore weakens the rod very much indeed. 

Constant of torsion of a rod or wire. The total force action across a complete 
section of a twisted rod is a torque T about the axis of the rod and the value of the 
torque is 

in which n is the slide modulus of the material of the rod or wire, R is the radius of 
the rod or wire, L is the length of the rod or wire, and 6 is the angle through which 

one end of the rod or wire is twisted. The nega- 
tive sign is written for the reason that the torque 
tends to reduce #,f that is, T and 6 are opposite 
in sign. This equation (57) shows that T is pro- 
portional to 6, and the proportionality factor 
irnR i /2L is called the constant of torsion of the 
wire or rod. If the constant of torsion of a rod 
or wire is determined by observing the angle of 
twist produced by a known torque, the slide 
modulus of the material may be calculated from 
equation (57). 

Proof of equation (57). Let Fig. 106 represent a 
Fig. 106. sectional view of the rod. Consider the narrow 

annulus of width Ar and radius r, as shown by 
the dotted lines. The force action per unit area across this annulus is rnOj L ac- 
cording to equation (56), and this force action is at right angles to r at each point of 

*The force actions between contiguous portions of a twisted rod are inferred from 
the character of the distortion at each point as represented in Fig. 105. These 
force actions may be made clearly evident by the use of two models, as follows: (a) 
A bundle of smooth square pine sticks bound together, and, if desired, turned to a 
cylindrical shape, shows sliding along vertical planes like q 3 , Fig. 105, when the 
bundle of sticks is twisted. (&) A brass tube with a slit along a helical line which is 
at each point inclined at an angle of 45 to the axis of the tube, is as strong as an 
uncut tube to withstand a twist in one direction (the faces of the slit push against 
each other normally), but the slit opens when the tube is twisted in the other 
direction. 

1"The reacting torque of the twisted rod is here referred to. 




ELASTICITY. (STATICS.) 213 

theannulus. Therefore the torque action, about the axis of the rod, of the force 
which acts across unit area of the annulus is rXrnd/ L, which, multiplied by the 
area 2Trr • Ar of the annulus, gives the total torque action AT across the annu- 
lus. That is 

. m 2ir r z nd . 
AT = — — - A r 



2ivnd C R ,, rcnR^d 
Jo ' 



r dr 



GENERAL EQUATIONS OF STRESS AND STRAIN. 

106. Principal stretches of a strain. A small spherical portion of a body always 
becomes an ellipsoid when the body is distorted.* A distortion which changes a 
sphere into an ellipsoid consists always of simple increase or decrease of linear di- 
mensions in three mutually perpendicular directions. These mutually perpendic- 
ular directions are called the axes of the strain, and the increase of length per unit 
length (I I L) in the directions of the respective axes are called the principal stretches 
of the strain. The principal stretches of a strain are represented by the letters /, 
g, and h. 

107. Principal pulls of a stress. Imagine a small plane area q, called a section, 
in the interior of a body under stress. The portions of the body on the two sides of 
this section exert on each other a definite force in a definite direction, and the force per 
unit area of the section is called the stress on the section. When the force is normal 
to the section, the stress on the section is called a pull, positive or negative as the 
case may be. When the force is parallel to the section, the stress on the section is 
called a tangential stress. 

The conditions of equilibrium of a small portion of a body under stress require* 
that at a point in the body there be three mutually perpendicular sections across 
which the force action is normal, or on which the stress is a pull. These three sec- 
tions are called the principal sections of the stress, the three lines perpendicular to 
them are called the axes of the stress, and the pulls on the three sections are called 
the principal pulls of the stress. These three pulls constitute the stress at the point 
and if these pulls are given the stress is completely specified. 

108. General equations of stress and strain. Let F be a longitudinal stress, 
that is, a simple pull, in the direction of the x-axis of reference. Such a stress causes 
a stretch aF in the direction of the x-axis, and a negative stretch — bF in the 
directions of the y and z-axes. Therefore, writing/', g' and h' for the three stretches 
due to the simple pull F, we have 

f'=oF 

g'=-bF (i) 

h'=-bF 

Similarly, let G be a longitudinal stress in the direction of the y-axis of reference, 
and let/'', g" , and h" be the three stretches, parallel to x, y, and z axes respectively, 
produced by G. Then we have: 

*See Elasticity, theory of, in Encyclopedia Britannica, oth Ed. 



214 ELEMENTS OF MECHANICS. 

f"=-bG 

i"=*G (ii) 

h"=-bG 

Similarly, let H be a longitudinal stress parallel to the z-axis of reference, and let 
/"'. g"', and h'" be the three stretches, parallel to the x, y, and z axes, respectively, 
produced by H. Then we have: 

f"=-bU 

g'"=-bH (iii) 

h'" = aH 

Experiment shows* that the stretch produced in any direction by a number of 
pulls acting together is equal to the sum of the stretches in that direction produced 
by the respective pulls acting separately. Therefore: ' . 

g = g'+g"+g'" (iv) 

h=h'+h"+h'" 

in which /, g, and h are the stretches, parallel to the x, y, and z axes, respectively, 
produced by the three pulls F, G, and H acting together. Therefore substituting 
the values of /', /", /"', g', g", g'", h', h", and W" from equations (i), (ii), and (iii) 
in equation (iv), and we have: 

f=aF-bG-bH 

g=-bF+aG-bH (58) 

h=-bF-bG+aH 

These equations give the strain (/, g, and h) which is produced in an isotropic 
elastic solid by any stress (F, G, and H), and it shows that an isotropic elastic solid 
has but two constants of elasticity a and b. In fact the quantity a is equal to 1/ E 
and the quantity b is equal to <ra, where <r is the value, /3' //3, of Poisson's ratio. 
See Art. 87. Starting with these relations, it is very easy to derive expressions for 
the bulk modulus B and for the slide modulus n of a substance in terms of E and <r 
by using equations (58); considering that F=G= H = p in the case of hydro- 
static pressure, and that F=-\-S, G=—S, and H = o in the case of a shearing 
stress. 

Problems. 
118. A helical spring is elongated by an amount of 1.2 inches 
when a 4-pound weight is hung upon it. How much additional 
elongation is produced by 1 pound additional weight? By two 

*In general, any effect which is proportional to a cause, may be resolved into parts 
which correspond to the parts of the cause. Thus a spring stretches in proportion 
to the stretching force. One kilogram produces, say, one centimeter elongation; 
two kilograms produce two centimeters elongation, which is one centimeter for each 
kilogram. See Art. 32. 



ELASTICITY. (STATICS.) 2 1 5 

pounds additional weight? By three pounds additional weight? 
By four pounds additional weight? Ans. 0.3 inch; 0.6 inch; 
0.9 inch; 1.2 inch. 

Note. Assume in this problem, and in those that follow, that the elastic limit is 
not exceeded. 

119. The middle of a long beam is depressed 2 inches by a load 
of 5,000 pounds. How much will it be depressed by a load of 
15,000 pounds? Ans. 6 inches. 

120. A long rod is fixed at one end, and a twisting force, or 
torque, of 100 pound-inches applied at the free end causes the free 
end to turn through an angle of io°. What torque would* be 
required to turn the free end of the rod through 26 ? Ans. 260 
pound-inches. 

121. A rod 2 inches in diameter and 20 feet long is stretched 
to a length of 20 feet and % inch by a force of 10,000 pounds- 
weight. What is the value of the longitudinal stress, and what is 
the value of the longitudinal strain? Ans. Stress 3,183 pounds 
per square inch; strain 0.001042. 

122. A rod 20 feet long and 1 inch in diameter is subjected to 
a pull of 20,000 pounds per unit of sectional area causing it to be 
lengthened to 20.02 feet, that is one part in a thousand, and caus- 
ing it to contract to a diameter of 0.99975 mcn > that is, 25 parts 
in one hundred thousand. What is the length and what is the 
diameter of the rod when it is subjected to a pull of 40,000 
pounds per unit sectional area? Ans. Length 20.04 feet, diameter 
0.9995 inch. 

123. A wire 200 inches long and 0.1 inch in diameter is pulled 
with a force of 150 pounds. The elongation produced is y& inch. 
What is the value of the stretch modulus of the material? Ans. 
7,640,000 pounds per square inch. 

124. A wire five feet long and 0.06 square inch sectional area 
is subjected to a stretching force of 300 pounds. The stretch 
modulus of the material is 28,000,000 pounds per square inch. 
What elongation is produced? Ans. 0.0107 inch. 

125. A steel beam is bent so that its middle line forms the arc 



2l6 ELEMENTS OF MECHANICS. 

of a circle 600 inches in radius. What is the elongation per unit 
length of a filament 2 inches from the middle line? Ans. 0.00333 . 

126. The stretch modulus of the steel of which the beam of the 
previous problem is made is 30,000,000 pounds per square inch. 
What is the pull (force per unit area of course) of a filament of 
the beam 2 inches from the middle line of the beam? Ans. 
100,000 pounds per square inch. 

127. What is the resilience of spring steel of which the elastic 
limit is 70,000 pounds per square inch and of which the stretch 
modulus is 30,000,000 pounds per square inch? Ans. 6.8 foot^ 
pounds per cubic inch. * . 

128. A cork y 2 inch in diameter is pushed with a force of 20 
pounds-weight into a bottle which is completely filled with water. 
What hydrostatic pressure is produced in the bottle? Neglect 
the friction of the cork against the glass neck of the bottle. Ans. 
101.8 pounds per square inch. 

129. A body subjected to hydrostatic pressure is decreased in 
length, in breadth, and in thickness by 5 parts in a thousand 
(initial) . By how many parts per thousand (initial) is the volume 
reduced? Ans. 15 parts per thousand. 

130. What is the value of the bulk modulus of water when 
2,000 cubic inches of water are reduced to 1,880 cubic inches 
by an hydrostatic pressure of 3,000 lbs. per square inch? Ans. 
50,000 pounds per square inch. 

131. Calculate the compressibility of water from the answer to 
the previous problem and explain its meaning. Ans. 0.00002 
square inch per pound. 

132. A bicycle pump is full of air at 15 pounds per square 
inch, length of stroke 12 inches; at what part of the stroke does 
air begin to enter the tire at 40 pounds per square inch above at- 
mospheric pressure? Assume the compression to take place with- 
out rise of temperature. Ans. When piston is 3.27 inches from 
the end of the cylinder. 

133. The clearance space behind the piston of an air compressor 
when the piston is at the end of its stroke is ■£$ of the volume 



ELASTICITY. (STATICS.) 217 

swept by piston during the stroke. What is the greatest pressure 
that can be produced in a compressed air reservoir by this com- 
pressor, the compression of the air in the cylinder being assumed 
to be without change of temperature? Ans. Fifty times the 
initial pressure. 

Note. As a matter of fact the air in an air compressor is heated very consider 
ably by the compression. 

134. The piston of an air pump is 0.0 1 inch from the bottom of 
the cylinder when it is at the end of its stroke, and the pressure of 
the air in the clearance space is then at atmospheric pressure. The 
length of stroke is 6 inches. What is the highest vacuum which 
can be produced by the pump ? Ans. 1 /600 of an atmosphere. 

135. A cubical piece of steel is shortened two parts in a thousand in one direction, 
lengthened two parts in a thousand in a direction at right angles to the first, and 
unchanged in dimension in the third direction, as represented in Fig. 101. What is 
the value of the angle of shear in degrees? Ans. 0.229 degree. 

136. A steel rod 120 inches long is fixed at one end and the other end is turned 
through 5 degrees of angle. Consider a small portion p of the rod at a distance of 1 
inch from the axis of the rod. Find the angle of shear <p of this small portion p of 
the metal. Ans. 0.0417 degree. 

137. The slide modulus of the steel used in the rod of problem 136 is 12 million 
pounds per square inch. Find the shearing stress in the small portion p of the rod. 
Ans. 8,735 pounds per square inch. 

138. A steel shaft 500 inches long and 3 inches in diameter transmitting 100 
horse-power is subject to a torque of 23,100 pound-inches of torque. The slide 
modulus of the material of the shaft is 12,000,000 pounds per square inch. Calcu- 
late the angle through which one end of the shaft is twisted relative to the other end. 
Ans. 0.12 1 radians or 6.94 degrees. 

139. The three stretches of a strain are +0.015, +0.025 and —0.025. What 
are the semi-axes of the ellipsoid into which a sphere 10 inches in radius is distorted 
by this strain? Strain supposed to be homogeneous. Ans. 10.15 inches, 10.25 
inches, 9.75 inches. 

140. A force of 250 pounds acts across a section of which the area is ]/i square 
inch. What is the value of the stress on the section? Ans. 1,000 pounds per square 
inch. 

141. A square rod 2X1^ inches is subjected to a tension of 75,000 pounds. 
What kind of stress acts across a section of the rod and what is its value? Ans. 
A normal stress of 25,000 pounds per square inch. 

142. Two long strips of metal are lapped and fastened by a single rivet of which 
the sectional area is two square inches. The two strips are subjected to a tension 
of 10,000 pounds. What kind of stress acts across the middle section of the rivet 
and what is the value of the stress? Ans. A tangential stress of 5,000 pounds per 
square inch. 



218 ELEMENTS OF MECHANICS. 

143. Derive the equation expressing the bulk modulus of a substance in terms of 
its stretch modulus and Poisson's ratio. The stretch modulus of steel is 30 million 
pounds per square inch, and Poisson's ratio for steel is 0.28. Find the value of the 
coefficient of compressibility and compare it with the value given in the table in 
Art. 99. One atmosphere is equal to 14.7 pounds per square inch. Ans. B = 
E/l3(i — 20-)]; CXio 6 = o.65; from table CXio 6 = o.68. 

144. Derive the equation expressing the slide modulus of a substance in terms of 
its stretch modulus and Poisson's ratio, and calculate the slide modulus of steel using 
the data given in problem 143. Ans. n= £/[2(i+cr)]; ra = n.72Xio 6 pounds per 
square inch. 



CHAPTER VIII. 

HYDROSTATICS. 

109. Pressure at a point in a fluid.* The force with which a 
fluid at rest pushes against an element of an exposed surface is 
at right angles to the element and proportional to the area of the 
element. The force per unit area is called the hydrostatic pres- 
sure or simply the pressure of the fluid at the place where the 
element of area is located and it is usually represented by the 
letter p. When the pressure has the same value throughout a 
fluid the pressure is said to be uniform, when the pressure varies 
from point to point in a fluid the pressure is said to be non-uni- 
form. When the pressure in a fluid is uniform the total force F 
acting on an exposed plane surface is 

F = pa (59) 

where a is the area of the surface, f 

Examples, (a) Steam pressure. The piston of a steam en- 
gine is pushed by a force equal to pa, where a is the area of 
the piston, and p is the pressure of the steam in the cylinder. 
Every part of the inside surface of a steam boiler is pushed out- 
wards by the steam. 

(b) Atmospheric pressure. The force with which the air 
pushes on the surfaces of bodies does not ordinarily appeal to our 
senses. It is shown however by the collapse of a thin-walled 
vessel when the inside pressure is reduced by pumping out the 
air. Atmospheric pressure is also strikingly shown by means of 
the apparatus known as the Magdeburg Hemispheres. This con- 
sists of two metal cups which fit together air tight and form a 
hollow vessel from which the air may be removed by pumping. 

*The term fluid includes liquids and gases, as explained in Art. 83. 
tSee Art. 97. 

219 



220 



ELEMENTS OF MECHANICS. 



The pressure of the outside air then holds the cups together and 
a considerable effort is required to separate them. This cele- 
brated experiment was devised by Otto von Guerike, the inven- 
tor of the air pump, and it was performed publicly in Magdeburg 
in 1654. 

(c) The hydrostatic press consists essentially of a strong cylin- 
der with a large plunger or piston, and a pump with a small pis- 
ton or plunger for forcing water into the large cylinder under 
high pressure. The great forging press at the Bethlehem Steel 
Works has two plungers each fifty inches in diameter, thus ex- 
posing a total of about 3,600 square inches of piston area to the 
water, which is forced into the cylinders of the press at a pres- 
sure of 8,000 pounds per square inch. This gives a total force 
of about 14,000 tons upon the two plungers. 

Pascal's principle. The force per unit area which is exerted 
by a fluid on an exposed surface at a given place in the fluid is 
independent of the direction of the surface. Thus the air pushes 
against an exposed surface with about 15 pounds of force per 
square inch whether the surface be horizontal or vertical or in- 
clined at any angle. 

110. The circumferential tension in the walls of a cylindrical 
pipe. The pressure of a fluid in a cylindrical pipe produces a 




Fig. 108. 



Fig. 109. 



tension in the material of the pipe. Consider a narrow band b, 
Fig. 107, of the material of a pipe, the width of the band being 
one inch. An end view of this band is shown in Fig. 108, and, 
since the band is one inch wide, each inch of its circumference is 



HYDROSTATICS. 221 

pushed outwards by a force equal to p pounds, where p is the 
steam or water pressure in pounds per square inch. Therefore, 
according to Art. 60, the circumferential tension in the band b is 
equal to rp pounds per inch of width, where r is the radius of 
the pipe in inches. 

It is instructive to establish this result from another point of 
view as follows: Imagine the cylindrical pipe to be half solid, as 
shown by the shaded area in Fig. 109, then, considering one inch 
of length of the pipe as before, the area of the flat surface ab is 
2r, the force acting on this flat face is 2rp , and this force is 
balanced by the two forces TT, so that the value of each force 
T is equal to rp. 

Since the circumferential tension in a cylindrical pipe is equal 
to rp, it is evident that a small pipe can withstand a much 
greater pressure than a large pipe, the thickness of the walls of 
the pipes being the same. 

Longitudinal tension in a boiler shell. Consider a cylindrical 
boiler of radius r. The area of each end of the boiler is irr 2 , 
the outward force exerted on each end of the boiler is irr 2 p, and 
this force produces lengthwise (longitudinal) tension in the boiler 
shell. Inasmuch as the width of the boiler shell which with- 
stands these endwise forces is equal to the circumference of the 
boiler (2irr), it is evident that the endwise tension is Trr 2 p/(2wr) 
or %pr units of force per unit width. 

111. Pressure in a liquid due to gravity. The pressure in a 
fluid under the action of gravity increases with the depth. If the 
density of the fluid is the same throughout, and this is approxi- 
mately the case in any liquid, then the pressure at a point distant 
x beneath the surface of the liquid exceeds the pressure at the 
surface by the amount 

p = xdg (60a) 

in which p is expressed in dynes per square centimeter when 
the density d of the liquid is expressed in grams per cubic centi- 
meter, the distance x in centimeters and the acceleration of 



222 ELEMENTS OF MECHANICS. 

gravity g in centimeters per second per second ; or p is expressed 
in poundals per square foot if the density of the liquid d is ex- 
pressed in pounds per cubic foot, the distance x in feet and 
the acceleration of gravity g in feet per second per second. 

The most useful form of the above equation is that which gives 
the pressure in pounds-weight per square foot, namely 

p = xd (fob) 

in which p is expressed in pounds per square foot, x is expressed 
in feet and d is the density of the liquid in pounds per cubic foot. 
Discussion of equation (66). The force with which a liquid 
pushes on an element of an exposed surface is independent of the 
direction of the surface element according to Pascal's principle. 
Therefore we may derive equation (60) by considering a horizon- 
tal surface a square feet in area exposed to the action of the liquid 



SMI 



surface of liquid 



Fig. no. Fig. in. 

as shown in Fig. no. The volume of the liquid directly above 
a is ax cubic feet, the mass of this portion of liquid is axd pounds 
where d is the density of the liquid in pounds per cubic foot, the 
force in poundals with which gravity pulls on this portion of the 
liquid is axdg, and therefore the total force with which this 
portion of liquid pushes down on the element a is equal to axdg 
poundals, so that the force per unit area is axdg divided by a, 
or xdg poundals per square foot. 

Equation (60) involves no consideration of the shape of the 
vessel which contains the liquid. As a matter of fact, the pres- 



HYDROSTATICS. 



223 



sure at a point in a liquid exceeds the pressure at the surface of 
the liquid by the amount xdg whatever the shape and size of 
the containing vessel may be. This may be made almost self- 
evident as follows: Given a point p, Fig. 111, at a distance x 
beneath the surface of a large body of liquid. Imagine a por- 
tion of the liquid AAA A, of any shape whatever, extending from 
p to the surface. The liquid surrounding the portion A AAA 
acts on AAAA exactly as a containing vessel of the same shape 
would act, and therefore the pressure of p is exactly what it 
would be if the portion AAAA were contained in such a vessel." 

112. The total force acting on a water gate and its point of application. When 
a plane surface of area a is exposed to the action of fluid under uniform pressure, 
the total force acting on the surface is pa and the point of application of this force 
is the center of figure ol the exposed plane surface. When, however, a plane surface 




Fig. 112a. 



Fig. 112&. 



is exposed to the action of the fluid in which the pressure is not uniform, the total 
force is, of course, not equal to pa, for p has different values at different parts of 
the surface, and the point of application of the total force is not at the center of 
figure of the exposed surface. The simplest case is that in which the water in a 
tank pushes against the rectangular side of the tank, or the case in which water 
pushes against a rectangular gate as shown in Fig. 112a. The pressure at the top 
of the gate is p' = 0.434*' pounds per square inch, the pressure at the bottom of the 
gate is £" = 0.434*", the average pressure over the whole gate is {p' r -\- p") / '2 or 
°-434 (x'+*")/ 2 pounds per square inch, and the total force F acting on the gate is 
equal to the product ot this average pressure and the area of the gate in square 
inches. 



224 



ELEMENTS OF MECHANICS. 



The point of application of the total force with which the water pushes on the 
gate is the point at which a single force F', Fig. 113, could be applied to balance the 
push of the water. This point is evidently below the center of the gate; in fact the 

distance X, Fig. 113, is 




X 



_ 2 / x" — X' \ 



(61) 



In the case of the side of a rect- 
angular tank, or in case of a dam 
(where x' equals zero) the distance 
from the surface of the water to 
the point of application of the 
total force which pushes on the 
side of the tank or against the 
dam is two thirds of the depth of 
the water. 

Proof of equation (61). The 
total force F, Fig. 113, is equal 
to the area of the gate w(x"—x') 
multiplied by the average pressure o.434(.r"+x')/2, w being the horizontal 
width of the gate; and the torque action of F about any conveniently chosen 
point is equal to the sum of the torque actions, about the same point, of the 
forces acting on the various elements of the surface of the gate. Consider a hori- 
zontal strip of the gate distant x beneath the surface of the water and of which the 
vertical breadth is dx. The force acting on this strip is o.434xXwdx, and the torque 
action of this force about a point at the surface of the water (lever arm x) is 0.434X 2 
Xwdx. Therefore the total torque action, about the chosen point, of the forces 
acting on the gate is equal to 



0.434W 



/ x 2 dx = 0.434WX K(*" 3 — *' 8 ) 



whence, placing this equal to the torque action XF[= XXo.434w(x' 
we have equation (61). 



*')X^], 



113. Measurement of pressure. The barometer. The bar- 
ometer consists of a glass tube T, Fig. 114, filled with mercury and 
inverted in an open vessel of mercury CC, the tube being of 
such length that an empty space V is left in which the pressure 
is zero.* The pressure in the tube at the level of the mercury 
in the open vessel is equal to atmospheric pressure, and it exceeds 
the pressure in the region V by the amount xdg according to 
equation (60). Therefore, since the pressure in V is zero, the 

*Even if the tube is filled with extreme care so as to exclude all of the air, mer- 
cury vapor will form in the region V and the pressure will not be exactly zero. 



HYDROSTATICS. 



225 



value of atmospheric pressure is equal to xdg. This expression 
gives the value of atmospheric pressure in dynes per square cen- 
timeter, x being in centimeters, d being the density of the mer- 
cury in grams per cubic centimeter, and g being the acceleration 
of gravity in centimeters per second per second. 

If the mercury is at some standard temperature, d is invariable; 
and if the barometer is used in a given locality, g is invariable; 
and under these conditions the distance x may be used as a measure 
of the pressure. In fact, atmospheric pressure is usually expressed 
in terms of the height the barometric column would have in 

millimeters or in inches if the mercury 
were at o° C. and if the value of the 
acceleration of gravity were 981.61 
cm. /sec 2 (its value at 45 ° north lati- 
• # tude at sea level) . To facilitate the 
•" • accurate use of the barometer in differ- 
ent localities and at different tempera- 
tures, tables* have been published, with 
. • the help of which the height of baro- 
metric column under standard condi- 
tions as to temperature and gravity 
may be easily found from its observed 
height under known conditions. 

114. Measurement of pressure. 
Manometers or pressure gauges. 
The barometer is used for the measure- 
ment of the pressure of the atmosphere. An instrument for 
measuring the difference between the pressure in a closed ves- 
sel and atmospheric pressure is called a manometer or a pressure 
gauge. 

The open tube manometer. When the pressure to be meas- 
ured is small, for example, when it is desired to measure the 




Fig. 114. 



*To be found in many laboratory reference books. For example, in Kohlrausch's 
Physical Measurements, and in Landolt and Bornstein's Physikalisch-Chemische 
Tabellen. 



226 



ELEMENTS OF MECHANICS. 






* 



_J. 



pressure of the gases at the base of a smoke-stack, or the pres- 
sure developed by a fan blower, the pressure is determined by 
measuring the height of water or mercury column which it will 
support. Thus Fig. 115 shows an open tube manometer ar- 
.... .-. Vi< fi . ranged for measuring the 
~J-5t '\*&*£-'$i~% P ress ure of the gas in city 
':if m \.\ mains. 

r.\'' The Bourdon gauge. 

.';■','■ The pressure gauge com- 
'-:'. monly used on steam boil- 
■•:•;.■ ers is usually of the type 
known as the Bourdon 
gauge, of which the essen- 
tial features are shown in 
Fig. 116. A very thin 
walled metal tube abc of flat 
elliptical section is closed 
at the end c, and the end a 
communicates through the 
tube tt with the steam boiler. The pressure inside of the tube 
abc tends to straighten it, and the movement of the end c ac- 
tuates a pointer which plays over a scale the divisions of which 
are determined by calibration, that is, by noting the position 
of the pointer for various known pressures. 

The gauge tester is a device for generating accurately known 
pressures which are communicated to a pressure gauge which is 
to be calibrated. It consists of a small metal chamber filled with 
oil. A plunger of known area a is forced into this chamber by 
a known weight, and the known pressure thus developed is com- 
municated to the gauge. 

115. Buoyant force of fluids. A fluid pushes upwards upon a 
body which is submerged in it and this upward force is called the 
buoyant force of the fluid. The buoyant force of a fluid upon a 
submerged body is equal to the weight of its volume of the fluid. 
This fact is called from its discoverer Archimedes' principle. 



Fig. 115. 



HYDROSTATICS. 



227 



The point of application of the buoyant force is the center of 
figure* of the submerged body, and it is called the center of 
buoyancy. 

The above statements may be made almost self-evident by 

the following considerations: Given a fluid at rest. Imagine a 

certain portion of this fluid of any size and shape. This portion 

is stationary, and therefore the surrounding fluid pushes upwards 

b 





to gauge 



Fig. 116. 



Fig. 117. 



upon it with a force which is equal to its weight, and the point 
of application of this upward force is the center of mass of the 
portion. But the surrounding fluid acts upon the given portion 
of fluid in exactly the same way that it would act upon a sub- 
merged body of the same size and shape. 

The principle of Archimedes is utilized in the ordinary method 
of finding the specific gravity of a body as follows : The body is 
weighed in the air and then it is suspended under water and 
weighed again. The difference is the weight of its volume of 
water, and the specific gravity of the body may then be calculated 
by dividing the weight (mass) of the body by the mass of its 
volume of water. 

When a body is weighed on a balance scale its weight (mass) is 
underestimated if it is more bulky than the weights that are used 

♦Center of mass of the body if the body is homogeneous. 



228 ELEMENTS OF MECHANICS. 

to balance it ; this is on account of the greater buoyant force ex- 
erted by the air on the body than on the weights. This error is 
often quite appreciable, and it must be allowed for in accurate 
weighing. 

A body which is partly submerged in a liquid is pushed up- 
wards by a force which is equal to the weight of the displaced 
volume of liquid. Therefore a floating body must displace its- 
weight of the liquid in which it floats* because the buoyant force 
must be just sufficient to balance the weight of the body. 

116. Equilibrium of a floating body.f A body is said to be 
in unstable equilibrium when the forces which act upon it tend to 
carry it farther and farther from its equilibrium position when it is 
displaced slightly therefrom. Thus a body standing vertically 
on a sharp point is in unstable equilibrium, the least displacement 
of the body in any direction causes it to fall over. 

A body is said to be in neutral equilibrium when the forces 
which act upon the body remain in equilibrium as the body 
moves. Thus a homogeneous sphere resting on a smooth hori- 
zontal table, and a balanced wheel supported on an axle are in 
neutral equilibrium. 

A body is said to be in stable equilibrium when the forces which 
act upon it tend to bring it back to its equilibrium position when 
it is displaced therefrom. Thus a weight fixed to the end of a 
spring, a pendulum hanging vertically downwards, and a block 
resting on a table are in stable equilibrium. 

A body is said to have a high degree of stability when a very 
considerable force is required.to displace it from its equilibrium 
position. Thus a broad sail-boat with its ballast placed low down 
in its hold is very stable, because a very considerable force is re- 
quired to turn the boat from its vertical position. 

Condition of equilibrium of a floating body. When a floating 
body is stationary, it is, of course, in equilibrium and the down- 
ward force of gravity must have the same line of action as the 

*Effects of capillary action are here ignored. 

fThis subject is treated in detail in works on naval architecture. 



HYDROSTATICS. 



229 



upward force of buoyancy, otherwise these two forces would have 
an unbalanced torque action and the body would not be in equi- 
librium. Therefore the center of a mass of a floating body and 
the center of figure of the submerged portion of the body {center 
of buoyancy) must lie in the same vertical line. 

The problem of determining the degree of stability of a floating 
body is greatly complicated by the change of shape of the sub- 
merged part of the body when the body is tilted to one side, 
and the shifting of the center of buoyancy which is due to this 




Fig. 119. 



change of shape. Therefore the simplest case is that of a floating 
body of which the submerged portion does not change its shape 
when the body is tilted to one side. 

Examples of simplest case. The submerged part of a floating 
sphere is the same in shape however the sphere be turned, and 
therefore the center of buoyancy does not move as the sphere is 
turned. If the center of mass of the sphere is at its geometrical 



230 



ELEMENTS OF MECHANICS. 



center we have a case of neutral equilibrium of floating; if the 
sphere is heavier on one side, it floats in stable equilibrium with 
its heavy side downwards, and in unstable equilibrium with its 
heavy side upwards. 

The most interesting simple example of equilibrium of floating 
is the hydrometer as shown in Figs. 118 and 119. The shape 



' I llr'll I ®" 6 ill I 11 ' 11 !!!-- 

"i^agME------- - - _-:3flSBSI:: 



Fig. 120. 

of the submerged portion is slightly altered when the hydrometer 
is tipped over, but the change of shape is nearly negligible and 
the center of buoyancy b is nearly fixed in position. 

Example of the general case. Consider two floats A and B, Figs. 120 and 121, 
connected rigidly together by a beam. This arrangement is similar to the style of 
boat called a catamaran, and when it is in equilibrium the center of mass m and the 
center of buoyancy b are located as shown in Fig. 120. When, however, the ar- 
rangement is tilted, as shown in Fig. 121, the center of buoyancy b shifts towards 
the lower side, while the center of mass m of course remains stationary. The 




arrangement behaves, for slight angles of tilting, as if its center of buoyancy were fixed in 
the line ma at the place B where the line ma is cut by the vertical line be in Fig. 121, 
because the line of action of the buoyant force passes through the point B for any small 
angle of tilling. The point B is called the metacenter of the float. 



HYDROSTATICS. 



231 



water-mark 



s-mark 



T 



117. The hydrometer. The common form of the hydrometer 
is a light glass float, weighted at one end with lead or mercury, 
and having a cylindrical glass stem at the other end, as shown in 
Fig. 118. This float sinks to different 
depths in liquids of different specific 
gravities, and upon the stem is a scale 
which indicates the specific gravity of 
the liquid in which the instrument is 
placed. 

The specific gravity scale. To con- 
struct a specific gravity scale on the 
stem of a hydrometer, the instrument 
is floated in water and the water-mark 
located; the instrument is then floated 
in a liquid of known specific gravity a, 
and the a-mark is located ; and then the 
distance I between the water-mark and 
the a-mark is measured. The scale is 
then determined by calculating the 
distance from the water-mark to each 

desired mark of the scale. Thus the distance d from the wa- 
ter-mark to the s-mark (s being a specific gravity, 1.10, 1.20, 
etc.) is given by the formula 

1 



Si-mark 



__J. 




I — 



d = l 



(62) 



This equation may be derived as follows: A floating body dis- 
places its weight of a liquid. The volume of water displaced by 
the instrument being taken as unity, the volume below the a-mark 
is 1 J a and the volume below the s-mark is 1 js inasmuch as these 
liquids are a times as heavy and s times as heavy as water 
respectively. Therefore the volume of the length I of the stem 
is (1 — 1 1 a) and the volume of the length d of the stem 



is 



232 ELEMENTS OF MECHANICS. 

(i — i/s), and, the stem being assumed cylindrical, the lengths 
I and d are proportional to these volumes. 

Beaume hydrometer scales. The specific gravity scale on a 
hydrometer is not a scale of equal parts and therefore the con- 
struction of the scale is tedious. On account of this fact a num- 
ber of schemes have been proposed for constructing hydrometers 
with arbitrary scales of equal parts. Of these scales those of 
Beaume are most extensively used. 

Beaume's scale for heavy liquids is constructed by locating the 
water-mark (near the top of the stem), and the mark to which 
the instrument sinks in a 15 per cent, solution of pure sodium 
chloride (common salt). The space between these marks is 
divided into 15 equal parts, and divisions of like size are con- 
tinued down the stem. These divisions are numbered down- 
wards from the water-mark. A liquid is said to have a specific 
gravity of 26 Beaume heavy when the hydrometer sinks in it to 
mark number twenty-six on the scale here described. 

Beaume's scale for light liquids is constructed by locating the 
mark to which the instrument sinks in a 10 per cent, solution of 
sodium chloride (near the bottom of the stem), and the water- 
mark. The space between these marks is divided into 10 equal 
parts, and divisions of like size are continued up the stem. These 
divisions are numbered upwards from the salt solution mark. A 
liquid is said to have a specific gravity of 17 Beaume light when 
the hydrometer sinks in it to mark number seventeen on the scale 
here described. 

CAPILLARY PHENOMENA OF LIQUIDS. 

118. Cohesion; adhesion. When a body is under stress, as for example a 
stretched wire, the tendency of the stress is to tear the contiguous parts of the body 
asunder. The forces which oppose this tendency and hold the contiguous parts 
of a body together are called the forces of cohesion. The forces which cause dis- 
similar substances to cling together are called the forces of adhesion. The dis- 
cussion of the elastic properties of solids is a discussion of their properties of co- 
hesion. The cohesion of water and the adhesion between water and glass are the 
forces which determine the curious behavior of water in a fine hair-like tube of 
glass, and the phenomena exhibited by liquids because of cohesion and adhesion are 
called capillary phenomena from the Latin word capillaris meaning hair. 



HYDROSTATICS. 



233 



119. Surface tension. On account of their cohesion, all liquids behave as if 
their free surfaces were stretched skins, that is, as if their free surfaces were under 
tension. Thus a drop of a liquid tends to assume a spherical shape on account of 
its surface tension. A mixture of water and alcohol may be made of the same 
density as olive oil, and a drop of olive oil suspended in such a mixture becomes 
perfectly spherical. 

Many curious phenomena* are produced by the variation of the surface ten- 
sion of a liquid with admixture of other liquids or with temperature. Thus a 
drop of kerosene spreads out in an ever widening layer on a clean water surface, 
on account of the fact that the tension of the clean water surface beyond the layer 
of oil is greater than the tension of the oily surface. A small shaving of camphor 
gum darts about in a very striking manner upon a clean water surface, on account of 
the fact that the camphor dissolves in the water more rapidly where the shaving 
happens to have a sharp projecting point, the water surface has a lessened tension 
where the camphor dissolves, and the greater tension on the opposite side pulls the 
shaving along. A thin layer of water on a horizontal glass plate draws itself away 
and leaves a dry spot where a drop of alcohol is let fall on the plate. A thin layer 
of lard on the bottom of a frying pan pulls itself away from the hotter parts of the 
pan and heaps itself up on the cooler parts, because of the greater surface tension 
of the cooler lard. 

120. Angles of contact. Capillary elevation and depression. The clean surface 
of a liquid always meets the clean walls of a containing vessel at a definite angle. 
Thus a clean surface of water turns upwards and meets a clean glass wall tangen- 
tially, and a clean surface of mercury turns downwards and meets a clean glass wall 
at an angle of 51 8'. 

Since a clean water surface turns upwards and meets a glass wall tangentially 
it is evident that the surface of water in a small glass tube must be concave as 
shown in Fig. 123, and the result is that the water is drawn up into the tube. On 





Fig. 123. 



Fig. 124. 



the other hand, the surface of mercury in a small glass tube is convex and the surface 
tension pulls the mercury down below the level of the surrounding mercury as 
shown in Fig. 124. 

*See the very interesting article capillary action in the Encyclopedia Britannica. 
This article also gives a comprehensive discussion of the theory of capillary action. 



234 



ELEMENTS OF MECHANICS. 



121. Measurement of surface tension of water. Let r be the radius of the bore 
of the glass tube in Fig. 123. Then the circumference 2irr is the width of the 
surface film of water at the point of tangency, and 2irrT is the total upward force 
due to the tension of the film, T being the tension per unit width. The volume of 
water in the tube above the level of the surrounding water is Trr 2 h, and the weight 
of this water is Tnr 2 hdg, where d is the density in grams per cubic centimeter and g 
is the acceleration of gravity. The weight of water in the tube being supported 
by the tension of the film, we have 



whence 



27TrT = TTr-, 

T 



rhdg 
2 



from which T may be calculated when r, d, and g are known and h observed. The 
surface tension of water is found in this way to be 81 dynes per centimeter breadth. 



Problems. 

145. Calculate the number of dynes per square centimeter in 
one pound- weight per square inch, taking the acceleration of 
gravity equal to 980 cm. /sec 2 . Ans. 68,900 dynes per square 
centimeter. 

146. Figure 146^? represents a hydrostatic press. The dis- 
tances a and b are equal to 6 inches and 6 feet respectively, the 





Fig. 146P. 

diameter of the pump plunger p is 1.5 inches and the diameter 
of the press plunger P is 24 inches. Find the total force on P 
due to a force of 100 pounds at F, neglecting friction. Ans. 
307,270 pounds-weight. 

147. Calculate the circumferential tension and the longitudinal 
tension in the cylindrical shell of a boiler due to a steam pressure 




HYDROSTATICS. 



235 



of 125 pounds per square inch, the diameter of the boiler being 
6 feet. Ans. 4,500 pounds-weight per inch and 2,250 pounds- 
weight per inch. 

148. Sheet steel 0.02 inch thick will safely stand a tension of 
200 pounds per inch of width. What is the greatest diameter 
of steel tube with 0.02 inch wall, which can safely withstand a 
pressure of 150 pounds per square inch? Ans. 2.66 inches. 

149. Calculate the pressure of the air in the caisson shown in 
Fig. 149^, the distance from the water level in the river to the 
water level in the caisson being 90 feet. Ans. 39.1 pounds per 
square inch. 

150. Oil and water are drawn up in two connecting tubes as 
shown in Fig. i$op. The height of the water column is 36 
inches, and the height of the oil column is 42 inches. What is the 




mm® 

m 




Fig. isop. 



Fig. 151^. 



ratio of the densities of oil and water (specific gravity of the oil)? 
Ans. 0.857. 

Note. The difference in pressure between the air in the upper part of the tube 
in Fig. 1 sop and the outside air is equal to x'd'g or to x"d"g, according to equation 
(60) where x' and x" are the heights of the respective liquid columns and d' and d" 
are the densities of the respective liquids. 

151. The pressure of illuminating gas in a gas holder (at the 
base of the holder) exceeds the pressure of the outside air at the 
same level by an amount which is equivalent to 5.08 centimeters 
of water. Find the difference between the gas and air pressures 
on top of a hill at a height h of 12,200 centimeters above the gas 



236 ELEMENTS OF MECHANICS. 

holder as shown in Fig. 151^. Ans. 9.956 grams-weight per 
square centimeter, or a pressure of 9.956 centimeters of 
water. 

Note. The density of air under ordinary conditions is 0.0012 gram per cubic 
centimeter, the density of illuminating gas is, say, 0.0008 gram per cubic centimeter 
and the density of water is one gram per cubic centimeter. In solving this problem 
use equation (60) and if it is desired to express pressures in grams weight per square 
centimeter omit the factor g. 

152. A vessel is filled with water and the hydrostatic pressure 
due to the water exerts a certain total force on the bottom of 
the vessel; sketch the form of the vessel for, which the total force 
on the bottom of the vessel may be (a) greater than, (b) equal to, 
and (c) less than the weight of the contained liquid. Ans. (a) 
Sketch a vessel in the form of a truncated cone or pyramid stand- 
ing on its large end ; (b) sketch a vessel in the form of a cylinder 
or prism; (c) sketch a vessel in the form of a truncated cone or 
pyramid standing on its small end. 

153. The density of mercury at o° C. is 13.5956 grams per 
cubic centimeter. Calculate the value in dynes per square centi- 
meter of standard atmospheric pressure, namely j6 cm. of mer- 
cury at o° C, the value of gravity being 980.61 cm. per second 
per second. Give the result in dynes per square centimeter. 
Ans. 1,012,900 dynes per square centimeter. 

154. The specific gravity of mercury is approximately 13.6. 
The pressure in pounds per square inch at a point x feet beneath 
pure water is p = 0.43 4.x. Find the value in pounds per square 
inch of one English standard atmosphere, namely, 30 inches of 
mercury. Ans. 14.75 pounds per square inch. 

155. Calculate the height of the homogeneous atmosphere; that 
is, assuming that the atmosphere has a uniform density of 
0.00129 grams per cubic centimeter throughout, calculate the 
depth which would produce standard atmospheric pressure. 
Ans. 8,012 meters or 4.98 miles. 

156. A piece of lead weighs 233.60 grams in air and 212.9 
grams in water at 20 C. What is the specific gravity and the 



HYDROSTATICS. 237 

density of lead at 20 C? Ans. Specific gravity 11.285; density 
11.265 grams per cubic centimeter. 

Note. The density of water at 20°C. is 0.998252 gram per cubic centimeter. 

157. A piece of glass weighs 260.7 grams in air and 153.8 
grams in water at 20 C. The same piece of glass weighs 92.2 
grams in dilute H 2 S0 4 at 20 C. What is the specific gravity of 
the H 2 SO, at 20 C? Ans. 1.576. 

158. A glass bulb weighs 75.405 grams when filled with air at 
standard temperature and pressure. It weighs 74.309 grams 
when the air is pumped out. It weighs 74.385 grams when 
filled with hydrogen at the same temperature and pressure. 
What is the specific gravity of hydrogen referred to air? Ans. 
0.06926. 

159. What is the net lifting capacity of a balloon containing 
400 cubic meters of hydrogen, its material weighing 250 kilo- 
grams? (Weight of a cubic meter of air is 1,200 grams; 
weight of a cubic meter of hydrogen is 90 grams.) Ans. 194 
kilograms. 

160. The distance along a hydrometer stem from the water 
mark to the mark to which the instrument sinks in kerosene 
(specific gravity 0.79) is 9.62 centimeters. Calculate the distance 
from the water mark to the marks to which the instrument would 
sink in a 20 % solution of alcohol, in a 40 % solution of alcohol, in 
a 60 % solution of alcohol, in an 80 % solution of alcohol, and in 
pure alcohol. The specific gravities of these solutions are as fol- 
lows: 20 % = 0.975 ; 40 % = 0.951 ; 60% = 0.913 ; 80 % = 0.863 
100% = 0.794. Ans. 0.93 centimeter, 1.86 centimeters, 3.44 
centimeters, 5.75 centimeters, 9.39 centimeters. 

161. The specific gravity of a 15 per cent, solution of sodium 
chloride at ordinary room temperature is 1.1115. Calculate the 
specific gravity corresponding to 26 Beaume (heavy). Ans. 
1. 21. 

Note. This problem is to be solved by using equation (62) in a way that will 
be apparent when it is considered that degrees Beaume represent distances along 
the hydrometer stem. 



238 ELEMENTS OF MECHANICS. 

162. The specific gravity of a 10 per cent, solution of sodium 
chloride at ordinary room temperature is 1.0734. Calculate the 
specific gravity corresponding to 20 Beaume (light). Ans. 0.936. 

163. Two rectangular boxes 12 inches X 12 inches X16 feet are fastened to 
two cross-beams like a catamaran, and the whole weighs 625 pounds. The distance 
apart from center to center of the two boxes is 6 feet. Find how far to one side 
the center of buoyancy shifts when the raft is tilted 5 about its longitudinal axis, 
find the approximate position of the metacenter, and find the torque tending to 
bring the raft into a horizontal position. Ans. The center of buoyancy shifts 2.52 
feet to one side; the metacenter is 28.9 feet above the center of mass of the float; 
and the torque tending to right the float is i,575 pound-feet. 

Note. For the sake of simplicity assume that the submerged part of each box 
remains rectangular. The metacenter is defined in terms of an infinitesimal angle 
of tilting, but its position for a 5 tilt may be determined approximately without the 
use of calculus in the case here considered. 



CHAPTER IX. 

HYDRAULICS. 

[Throughout this chapter the following units are used: feet, 
feet per second, square feet, cubic feet, pounds (mass), pounds 
per cubic foot (density), pounds (force), pounds per square foot 
(pressure), and foot-pounds (energy). The factor g is equal to 
32.2 which is the acceleration in feet per second per second of a 
one pound body when acted upon by an unbalanced force of one 
pound-weight.] 

122. Limitations of this chapter. Hydraulics, in the general 
sense in which the term is here used, is the study of liquids and 
gases in motion; and the phenomena which are presented in this 
branch of physics are excessively complicated. Even the appar- 
ently steady flow of a great river through a smooth sandy chan- 
nel is an endlessly intricate combination of boiling and whirling 
motion; and the jet of spray from a hydrant, or the burst of 
steam from the safety-valve of a locomotive, what is to be said of 
such things as these? Or let one consider the fitful motion of 
the wind as indicated by the swaying of trees and as actually vis- 
ible in driven clouds of dust and smoke, or the sweep of the 
flames in a conflagration! These are actual examples of fluids 
in motion, and they are indescribably, infinitely* complicated. 
The finer details of such phenomena, however, are devoid of 
practical significance, indeed they present but little that is suffi- 
ciently definite even to be described. 

♦Everyone concedes the idea of infinity which is based upon abstract numerals 
(one, two, three, four and so on ad infiniluml), and the idea of infinity which is 
based on the notion of a straight line; but most men are wholly concerned with the 
humanly significant and persistent phases of the material world, their perception 
does penetrate into the substratum of utterly confused and erratic action which 
underlies every physical phenomenon, and they balk at the suggestion that the 
phenomena of fluid motion, for example, are infinitely complicated. Surely the 
abstract idea of infinity is as nothing compared with the intimation of infinity that 
comes from things that are seen and felt. 

239 



240 ELEMENTS OF MECHANICS. 

The science of hydraulics is based on ideas which refer to general 
aspects of fluid motion, like a sailor's idea of a ten-knot wind*; and, 
indeed, the engineer is concerned chiefly with what may be called 
average effects such as the time required to draw a pail of water 
from a hydrant, the loss of pressure in a line of pipe between a 
pump and a fire nozzle, or the force exerted by a water jet on the 
buckets of a water wheel. These are called average effects be- 
cause they are never perfectly steady but always subject to per- 
ceptible fluctuations of an erratic character, and to think of any 
of these effects as having a definite value is, of course, to think of 
its average value under the given conditions, f The extent to 
which the practical science of hydraulics is limited is evident from 
the following outline of the ideal types of flow upon which nearly 
the whole of the science is based. 

Permanent and varying states of flow. When a hydrant is 
suddenly opened, it takes an appreciable time for the flow of 
water to become steady. During this time (a) the velocity at each 
point of the stream is increasing and perhaps changing in direc- 
tion also. After a short time, however, the flow becomes fully 
established and then (Jb) the velocity at each point in the stream re- 
mains unchanged in magnitude and direction. % The motion (a) 
is called a varying state of flow, and the motion (b) is called a 
permanent state of flow. Most of the following discussion applies 
to permanent states of flow, indeed there are but few cases in 
which it is important to consider varying states of flow. 

The idea of simple. flow. Stream lines. The idea of simple 
flow applies both to permanent and to varying states of flow, but 
it is sufficient to explain the idea in its application to permanent 
flow only. When water flows steadily through a pipe, the motion 
is always more or less complicated by continually changing 
eddies, the water at a given point does not continue to move in a 

*See page 4. 

tSee two brief articles by W. S. Franklin, Transactions of American Institute of 
Electrical Engineers, Vol. XX, pages 285-286; and Science, Vol. XIV, pages 496- 
497, September 27, 1901. 

JAssuming the stream to be free from turbulence. See the following definition 
of simple flow. 



HYDRAULICS. 



241 



fixed direction at a constant velocity; nevertheless, it is conve- 
nient to treat the motion as if the velocity of the water were in 
a fixed direction and of constant magnitude at each point. Such a 
motion is called a simple flow . In the case of a simple flow, a line 




can be imagined to be drawn through the fluid so as to be at 
each point in the direction of the flow at that point. Such 
a line is called a stream line. Thus the fine lines in Fig. 125 
are stream lines representing a simple flow of water through a 
contracted part of a pipe. To apply the idea of simple flow to 
an actual case of fluid motion is the same thing as to consider 




Fig. 126. 

the average character of the motion during a fairly long interval of 
time. 

Lamellar flow. Even though the motion of water in a pipe 
may be approximately a simple flow, the velocity may not be the 
same at every point in a given cross-section of the pipe, that is, 
the velocity may not be the same at every part of the layer ab, 
Fig. 126; in fact the water near the walls always moves slower 
than the water near the center of the pipe; nevertheless, it is con- 
venient in many cases to treat the motion as if the velocity were 
the same at every point in any layer like ab, Fig. 126. Such an 
ideal flow is called a lamellar flow, because in such a flow the 
fluid in any layer or lamella ab would later be found in the layer 
17 



242 



ELEMENTS OF MECHANICS. 



cd, and still later in the layer ef. To apply the idea of lamellar 
flow to an actual case of fluid motion is the same thing as to con- 
sider the average velocity over the entire cross-section of a stream. 

Rotational and irrotational flow. In certain cases of fluid motion each particle of 
the fluid, if suddenly solidified, would be found to be rotating at a definite angular 
velocity about a definite axis; such fluid motion is called rotational motion or vortex 
motion. Thus the whirling motion of the water in an emptying sink is vortex motion. 
In other cases of fluid motion the particles of the fluid are not rotating; this kind of 
fluid motion is called irrotational motion. Some of the important practical aspects 
of vortex motion are discussed in the Encyclopedia Britannica article Hydrome- 
chanics, Part III., Hydraulics, section 30, 31, 103 and 190. 

In irrotational fluid motion the velocity can be represented as a potential gra- 
dient, whereas in rotational fluid motion the velocity cannot be represented as a 
potential gradient. See Art. 18, space variation of vectors. 

123. Some actual phenomena of fluid motion. The following 
treatment of fluid motion is so largely based upon the idea of 
simple lamellar flow that in pursuing the discussion we will be 
carried far away from any consideration of the details of actual 



v *afrj 




Fig. 127. 

fluid motion, and, although many of these details are essentially 
erratic, still there are a few details which are definitely typical. 
The water hammer. The most striking phenomenon that is 
associated with a varying state of fluid motion is the effect pro- 
duced when an open hydrant is suddenly closed; the momen- 
tum of the water in the pipe causes the water to exert on the 
suddenly closed valve a momentary force very much like a 
hammer blow. This momentary force is often excessively large 
in value and a valve which is closed suddenly should be protected 



HYDRAULICS. 



243 



by an air cushion as shown in Fig. 127. The sharp rattling 
noise which is occasionally produced in steam pipes is due to the 
"water hammer." A column of condensed water is driven along 
the pipe by the steam, the cooler steam ahead of the column 
condenses, and the column of water hammers against the end of 
the pipe or against a stationary body of water in the pipe. 

The hydraulic ram consists of a valve A, Fig. 128, arranged 
to automatically open and close the end of a long pipe PP. 




Fig. 128. 



When the valve opens, the water from the dam starts to flow, 
this flow lifts the valve A thus suddenly closing the end of the 
pipe PP, and the momentum of the water in PP generates a 
momentary pressure which lifts the valve B and forces a small 
quantity of water to a high storage tank. The valve A then 
falls, and the action is repeated. The automatic opening of the 
valve A is due to the recoil of the water in the pipe PP as fol- 
lows: At the moment when the water in PP is brought to rest 
in forcing water into the storage tank, the pressure at the end of 
PP is of course still excessive and the water near the end of PP 
is compressed. This compression then relieves itself by starting 
a momentary backward flow, or recoil, of the water in PP, 
and this recoil is followed by a momentary decrease of pressure 
sufficient to allow the valve A to drop. 

The sensitive flame. When a fluid flows through a fairly 



244 ELEMENTS OF MECHANICS. 

smooth walled pipe, the motion approximates very closely to a 
simple flow if the velocity is not excessive ; but when the velocity 
is increased the motion tends to become more and more turbulent 
(full of eddies) , and in many cases there is a fairly definite velocity 
at which the motion suddenly becomes very turbulent. This is 
shown by watching the movement of "sawdust- water" through 
a large glass tube. At low velocities the particles of sawdust 
move in fairly straight paths, but as the velocity of flow is in- 
creased the particles begin to gyrate with considerable violence 
when a certain critical velocity is reached. 

This sudden increase of turbulence is illustrated by the familiar 
behavior of a gas flame. When the gas is turned on more and 
more the flame remains fairly steady until the velocity of the 
flowing gas reaches a certain critical value and then the flame 
suddenly becomes rough and unsteady. When the flame is on 
the verge of becoming unsteady it is sometimes very sentitive; 
the least hissing noise causes it to become turbulent. An ex- 
tremely sensitive flame may be obtained by burning ordinary 
illuminating gas from a smooth circular nozzle made by drawing 
a glass tube down to the desired size (about T / 2 millimeter to I 
millimeter diameter of opening) . Generally, several nozzles must 
be tried before one is found that is suited to the gas pressure that 
is available. 

Vortex rings. When a fluid is at rest, mixing takes place 
only by the very slow process of diffusion, and when a fluid is 
in turbulent motion the mixing of the different parts of the fluid 
takes place very rapidly on account of the eddies which constitute 
the turbulence. The slowness of mixing of a smoothly flowing 
fluid, however, is illustrated by the smooth gas flame and by the 
threads of smoke that rise from the end of a cigar. Such a 
stream of fluid flowing smoothly through a large body of fluid at 
rest tends always to break up into what are called vortex rings. 
Thus a fine jet of colored water entering at the top of a large 
vessel of clear water and streaming towards the bottom, breaks 
up into lings which spread out wider and wider as they move 



HYDRAULICS. 



245 



downwards, each ring preserving its identity (not mixing with the 
clear water). The most interesting example of the formation of 
vortex rings is the familiar case of the formation of smoke rings 
when smoke issues as a moderate puff from an orifice into the 
air. Of course the smoke only serves to make the rings visible, 
and a candle can be blown out by an invisible vortex ring of air 
projected across a large room from an orifice in a box by striking 
a flexible diaphragm which is stretched like a drum head across 
the back of the box. 

Cyclonic movements. When water flows out of a hole in the 
bottom of a bowl a whirlpool generally forms above the hole. 
The formation of this whirlpool depends upon the existence of a 
slow rotatory motion of the water in the bowl, which rotatory 
motion is greatly increased when the water flows toward the hole 
as a center. This increase of velocity as a rotating particle is 
made to approach the center may be explained with the help of 
Fig. 129a as follows: A ball B is twirled on a string 5 which passes 



B 

r 



B 

f 



{"6 



front view 



side view 



Fig. 129a. 



through a tube TT, and after the ball is set in motion the string 
is pulled by taking hold of it at P, thus causing the ball to move 
along the dotted spiral. The velocity of the ball is increased as 
it moves towards the axis; the work done in pulling the string 
increases the kinetic energy of the ball. 



246 



ELEMENTS OF MECHANICS. 




Fig. 129&. 



Figure 1296 represents a whirlpool of water in a deep circular 
bowl. In this case the values of the hydrostatic pressure at 
the points a, b and c are proportional to the lengths of the re- 
spective dotted lines, that is to say, the 
hydrostatic pressure decreases towards 
the axis of the bowl, and it is this de- 
crease of hydrostatic pressure which causes 
an increase of velocity of a given particle 
of water as it moves towards the axis of the 
bowl,* as explained in Arts. 126, 127 and 
128. ' • 

The rotation of the earth on its axis 
involves a slow motion of turning of one's 
horizon about a vertical axis (except at the 
equator). When the warm air near the earth's surface starts to 
flow upwards at a given point, a chimney-like effect is produced 
by the rising column of warm air, the lower layers of warm air 
flow towards this "chimney" from all sides, and the slow turn- 
ing motion of the horizon becomes very greatly exaggerated in a 
more or less violent whirl at the "chimney" which is the center 
of the storm. The cyclone is a storm movement of this kind cov- 
ering hundreds of thousands of square miles of territory with a 
central chimney hundreds of miles in diameter, the tornado is a 
storm movement of this kind covering only a few square miles 
of territory with a central chimney seldom more than a thousand 
yards in diameter. The whirling motion near the center of a 
tornado is often excessively violent. 

124. Rate of discharge of a stream. The volume of water 
which is delivered per second by a stream is called the discharge 
rate of the stream. Thus the mean discharge rate of the Niagara 
River is 300,000 cubic feet per second. The rate of discharge of a 
stream is equal to the product of the avergae velocity, v, of the stream 

*This increase of rotatory motion due to the movement of a portion of a rotating 
system towards the axis of rotation is an example of the constancy of spin momen- 
tum of a system on which no outside force (torque) acts, as explained in Art. 62. 



HYDRAULICS. 247 

and the sectional area, a, of the stream. For example, let PP, 
Fig. 130, be the end of a pipe out of which water is flowing, and 
let us assume that the velocity of flow has the same value v over 
the entire section of the stream (lamellar flow), then the water 
which flows out of the end of the pipe in t seconds would make 
a cylinder or prism of length vt, and of sectional area a, as 
indicated in the figure, and the volume of this water is therefore 
avt. Dividing this volume by the time t gives the discharge rate 
wo. 

Variation of velocity with sectional area of a steady stream. 
Consider a simple flow of water through a pipe as indicated by 
the stream lines in Fig. 125. Let a' and a" be the cross-sectional 
areas of the stream at any two points P' and P" and let v' ', and 




vi _H 

Fig. 130. 

v" be the average velocities of the stream at P' and P" respect- 
ively. Then a'v' is the volume of water which passes the point 
P' per second, and a"v" is the volume of water which passes 
the point P" per second; and, therefore, since the same amount 
of water must pass each point per second, we have 

cV = a"v" (63) 

that is, the product av has the same value all along the pipe, 
so that v is large where a is small, and v is small where a is large. 

Equation (63) applies only to a fluid which is approximately incompressible 
like water or any other liquid. In such a case a'v' is the amount of water per 
second entering one end of a pipe and a"v" is the amount of water per second 
flowing out of the other end of the pipe, and these two expressions must be equal 
to each other. If, however, the fluid is compressible like a gas, then equation (63) 

becomes 

a'v'd' = a"v"d" (64) 



248 ELEMENTS OF MECHANICS. 

where a' is the sectional area of the steady stream of gas at one place, v' is the 
average velocity of the stream at that place, d' is the density of the gas at that place, 
and a", v" and d" are the cross sectional area, the velocity of the stream and the 
density of the gas at another part of the stream. 

125. The ideal frictionless incompressible fluid. When a jet 
of water issues from a tank, there is a certain relation between the 
velocity of the jet and the difference in pressure inside and out- 
side of the tank. When there are variations of the velocity 
of flow of water along a pipe due to enlargements or contrac- 
tions of the pipe [see equation (63)], the pressure decreases 
wherever the velocity increases and vice versa. These mutually 
dependent changes of velocity and pressure are always compli- 
cated by friction, and by the variations of the density of the fluid 
due to the variations of pressure ; and in order to gain the simplest 
possible idea of these mutually dependent changes of velocity 
and pressure the conception of the frictionless incompressible fluid 
is very useful. 

When the water in a pail is set in motion by stirring, it soon 
comes to rest when left to itself. A fluid which would continue 
to move indefinitely after stirring would be called a frictionless 
fluid. 

When a moving fluid is brought to rest by friction, the kinetic 
energy of the moving fluid is converted into heat and lost. Such 
a loss of energy would not take place in a frictionless fluid, 
and therefore the total energy (kinetic energy plus potential 
energy) of a frictionless fluid would be constant. This prin- 
ciple of the constancy of total energy is the basis of the following 
discussion of the flow of the ideal frictionless fluid. The follow- 
ing discussion applies to fluids which are not only frictionless 
but also incompressible. In fact, ordinary liquids are nearly 
incompressible. 

126. Energy of a liquid, (a) Potential energy per unit of volume. 
Work must be done to pump a liquid into a region under pres- 
sure, the amount of work done in pumping one unit of volume 
of the liquid is the potential energy per unit of volume of the 



HYDRAULICS. 



249 



liquid in the high pressure region, and it is equal to the pressure. 
That is W = p (65) 

In this equation W and p may both be expressed in c.g.s. units 
or in the units enumerated at the head of this chapter. 

Proof of equation (65). Let CC, Fig. 131, be the cylinder 
of a pump which is used to pump liquid into a tank under a 
pressure of p pounds per square foot, and let the area of the 
piston be a square feet. Then the force required to move the 
piston (ignoring friction) is ap pounds, and the work done in 
moving the piston through a distance of I feet is apl foot- 
pounds. But al is the volume of water pushed into the tank 




Fig. 131. 

by the movement of the piston, and therefore, dividing apl by 
al gives the work in foot-pounds required to push one cubic 
foot of water into the tank. 

When a stream of liquid moves in a horizontal plane, the 
gravity pull of the earth does no work on the liquid; but when 
a stream flows through an inclined pipe the gravity pull of the 
earth does work (positively or negatively) on the liquid and it is 
necessary therefore in this case to consider the energy of altitude 
as a part of the potential energy of the liquid. In fact, one cubic 
foot of liquid of which the weight is d pounds has potential 
energy equal to hd foot-pounds when it is at a height of h feet 
above a chosen reference level, so that the potential energy 
of the liquid per cubic foot is 

W' = p + hd (66) 



250 



ELEMENTS OF MECHANICS. 



In the discussion in Arts. 128 and 129 the effects of gravity 
are ignored, that is to say, the pipe or stream is supposed to be 
horizontal. If it is desired to consider the effects of gravity, 
p + hd can be substituted for p in the equations of Art. 129. 

(b) Kinetic energy. Let v be the velocity in feet per second 
of a moving liquid, and let d be the mass of one cubic foot of the 
liquid in pounds (d is the density of the liquid). Then the kinetic 
energy of one cubic foot of the liquid in foot-pounds is 



W 



2g 



dv 2 



(67) 



Consider a tank, Fig. 



according to equation (27) of chapter V. 

127. Efflux of a liquid from a tank. 

132, containing a liquid of which the density is d pounds per 

cubic foot. Let 00 be an 

ran 

ni — 1 

i 




orifice from which the liquid 
issues as a jet at a velocity 
v feet per second to be 
determined. Let p pounds 
per square foot be the 
pressure in the tank at the 
level of the orifice, and let 
p' be the outside pressure 
(atmospheric pressure). In 
the tank, where the veloc- 
ity of the liquid is inappreciable, the total energy of the liquid 
per unit of volume is the potential energy p [equation (65)]. 
In the jet the total energy per unit volume is p' + i/2g X dv 2 
[equations (65) and (6j)]. As a portion of the liquid moves 
from the tank into the jet its total energy would remain un- 
changed if it were frictionless, so that we would have 



Fig. 132. 



p = p'+- dv 2 

* * ^ 2g 



whence 



v = 



2g(p - P') 



d 



(68) 



HYDRAULICS. 25 1 

This equation expresses the velocity of efflux of a frictionless 
incompressible fluid. The effect of friction is to decrease v, and 
the effect of compressibility is to increase v. For ordinary liquids 
the effect of friction is the greater, and equation (68) gives too 
large a value for v. For gases the effect of compressibility is the 
greater, and equation (68) gives too small a value for v. 

Torricelli's theroem. The velocity of efflux of a frictionless 
liquid is equal to the velocity a body would gain in falling freely 
through the distance x of Fig. 132. The pressure-difference 
p — p' is equal to xd, according to equation (60)6 of chapter 
VIII, so that, substituting xd for p '— p' in equation (68), we 
have 

V = V2gX 

and this is the velocity gained by a body in falling through the 
distance x, according to Art. 35. 

The kinetic energy, per cubic foot, of the liquid in the jet is 
equal to the work that would be done by gravity on one cubic 
foot of water in falling from the surface of the liquid in the tank 
to the level of the orifice. 

128. Bernoulli's principle. Consider a stream of water flowing 
through a pipe. Wherever the pipe is small in section the veloc- 
ity of the water is great, and wherever the pipe is large in sec- 
tion the velocity of the water is small, according to equation 

(63). 

Let p be the pressure of the water at a point in a pipe, and v 
the velocity of the water at that point. Then the potential 
energy of the water per cubic foot is p, and the kinetic energy of 

the water per cubic foot is — dv 2 according to Art. 126, and if the 
liquid is frictionless so that no energy can be lost (converted into 
heat) , then the total energy p -f — dv 2 must be constant every- 
where along the stream. Therefore, where the velocity is great, 
the pressure must be small, and where the velocity is small, the 



252 



ELEMENTS OF MECHANICS. 



pressure must be great. This relation was first established by 

John Bernoulli and it is known as Bernoulli's principle. 

Examples, (a) The disk paradox. Figure 133 represents a 

short piece of tube T ending in a flat disk DD, and dd is a light 

metal disk which is prevented from moving sidewise by a pin 
which projects into the end of the tube 
T. If one blows hard through the tube T, 
the disk dd is held tight against DD be- 
cause of the low pressure in the very 
greatly contracted portion of the air stream 
between the disks. In fact, the pressure of 
the air in the region between the disks is 
less than atmospheric pressure, and it in- 
creases towards the edge of the disks as 

the velocity of the air stream diminishes (and the sectional area 

of the stream increases). 

(b) The jet pump. The essential features of the jet jump are 

shown in Fig. 134. Water from a fairly high-pressure supply H 





v/^/^^//////////////////m 



Fig. 134. 



enters a narrow throat, the low pressure in the throat sucks water 
from A A , and the water from H, together with the water from 



HYDRAULICS. 



253 



A A, is discharged into the reservoir R. This type of pump is 
frequently used for pumping water out of cellars, and it is exten- 
sively used as an air pump in chemical laboratories. 

The steam injector is a jet pump, and its paradoxical action in 
pumping water into a boiler at the same pressure* as the steam 
supply depends upon the low density of the steam and upon 
the fact that the steam is condensed in the injector. The low- 
density steam acquires a very high velocity in flowing out of 
the boiler, according to equation (68), and the velocity which is 
imparted to the water in the injector (including the water from 
the condensed steam) is sufficient to carry the water back into 
the boiler. 

(c) The volume of water discharged per second from a given 
sized orifice 00, Fig. 135, is greatly increased by the flaring tube 




Fig. 135- 



AB. The rate of discharge of a frictionless fluid would depend 
only upon the size of the open end B of the tube, the contraction 
at A would have no effect. In the case of an actual fluid, the 
effect of the contraction at A is to increase the friction consider- 
ably and thus reduce the discharge rate below what it would be 
if the tube at A were as large as at B. 

(d) The curved flight of a base-ball. Figure 136a shows a stream 
of air flowing past a stationary ball. In this case the velocity of 
the air is the same at the points a and b, and therefore, the pressure 

*Indeed the injector can pump water into a region in which the pressure is 
greater than the pressure of the supply stream. 



254 



ELEMENTS OF MECHANICS. 



of the air is the same on both sides of the ball. If, however, the 
ball is rotating and if its surface is slightly rough, the surrounding 
air will be set into a whirl, and in this case the stream lines of the 




Fig. 136a. 

air flowing past the ball will be some what as shown in Fig. 13 6b. 
The velocity at each point in this figure is the resultant of two 
velocities, namely, the velocity due to the stream alone (as shown 
in Fig. 136a), and the velocity due to the whirl alone. In Fig. 
136& the velocity of the air is greater in the region b than it is 
in the region a, and according to Bernoulli's principle the pressure 




Fig. 136&. 

of the air is less in the region b than it is in the region a. There- 
fore the excess of pressure in the region a pushes the ball from a 
towards b. 

Figures 136a and 1366 show a stream of air flowing past a sta- 
tionary ball. The same effects, however, are produced when the 



HYDRAULICS. 



255 



ball moves through a stationary body of air in the direction 
from right .to left in either figure. 





Fig. 137. 

(e) An interesting phenomenon dependent upon Bernoulli's 
principle is that two ships steaming along side by side attract 
each other and are in danger of being drawn together. Thus 




Fig. 138. 

when a squadron of warships is maneuvering, it is dangerous for 
one ship to attempt to pass close by another. Figure 137 repre- 
sents two stationary boats A and B with a stream of water flow- 



256 



ELEMENTS OF MECHANICS. 



ing past them. The velocity of the water is much greater in 
the region between the ships than it is on either side a.s indicated 
by the stream lines, therefore the pressure in the region between 
the ships is less* than the pressure in the regions aa, and therefore 
the ships are pushed towards each other. Figure 137 represents 
a stream of water flowing past two stationary ships; the effect, 
however, is the same if the two ships are moving through a large 
body of still water. 

(/) Figure 138 shows the approximate trend of the lines of 
flow of a stream in the neighborhood of a flat plate PP. The 
velocity of the stream is very great at the point a and small at 
the point b. Therefore the pressure of the fluid is great at b 
and small at a, and this difference in pressure tends to cause the 
plate to rotate in the direction of the curved arrows, thus bringing 
the plate into a position with its plane at right angles to the 
stream. 

129. Diminution of pressure in a throat. A contracted por- 
tion of a pipe is called a throat. When a fluid flows through a 
A 




pipe in which there is a throat the velocity of the fluid in the throat 
is greater than it is in the larger portions of the pipe, and there- 
fore the pressure of the fluid in the throat is less than it is in the 
larger portions of the pipe. Let Fig. 139 represent a pipe with 
a throat; let a' be the cross-sectional area of the pipe at A, and 
let p' and v ' be the pressure and velocity respectively of the fluid 
at A ; and let a", p" and v" be the corresponding quantities at 
B. Then, if the fluid is incompressible, we have 

a'v' = a"v" (i) 

*The water level between the two boats is lower than the normal water level. 



HYDRAULICS. 257 

according to equation (63), and if the fluid is also frictionless we 
have 

*'+-<ft/W'-f -dv" 2 (ii) 

Therefore, substituting the value of z;" from (i) in (ii) and solving 
for £' — p'\ we have 

where pi — />" is the diminution of pressure in the throat. 

The diminution of pressure in a throat is explained directly 
from Newton's second law of motion as follows: Consider a 
particle of liquid at A, Fig. 139. This particle gains velocity as 
it approaches B, and loses velocity again as it approaches C. 
Therefore, an unbalanced force must be pushing the particle for- 
wards as it passes from A to B, that is, the pressure in the region 
behind the particle is greater than the pressure in the region 
ahead of the particle ; and an unbalanced force must be opposing 
the motion of the particle as it passes from B to C, that is, the 
pressure ahead of the particle is greater than the pressure behind 
it. 

The diminution of pressure at a throat in a pipe is, of course, 
an example of Bernoulli's principle, and it is exemplified by the 
disk paradox which is described in Art. 128. 

The Venturi water meter consists of a throat inserted in a 
water pipe through which the water to be measured flows. The 
diminution of pressure p' — p" [see equation (69)] is measured, 
and, since the cross-sectional areas a' and a" are known, the 
velocity v' and the rate of discharge a'v' can be calculated 
from the measured value of p' — p".. 

130. Reaction of a water jet. Force of impact of a jet. Figure 
140 represents a tank containing water at a pressure p (in excess 
of outside pressure). The tank has an orifice of area a, and the 
orifice is closed by a plug P. The force acting on the plug is 
equal to pa, and the total force pushing on the side A A of the 
18 



258 



ELEMENTS OF MECHANICS. 



tank is equal to total force pushing on the side BB including 
the force acting on the plug. Therefore, it would seem that an 
unbalanced force equal to pa would push the tank towards the 
left in Fig. 140 if the plug were removed; but when the plug 
is removed there is a reduction of pressure in the neighborhood 
of the orifice as indicated by the very small arrows in Fig. 141, 
so that the unbalanced force which pushes the tank towards the 
left in Fig. 141 is much greater than pa, it is in fact equal to 




Fig. 140. 



Fig. 141. 



ipa on the following assumptions, namely, (a) that the velocity 
of efflux is that of an ideal incompressible fluid, and (b) that the 
jet issues as a parallel stream of the same size as the orifice. 

It is difficult, however, to show that the reaction of the jet is 
2pa by considering the change of pressure inside of the tank due 
to the existence of the jet, but the reaction can be evaluated in a 
comparatively simple manner by considering the force which 
must act on the outflowing water to set it in motion. In one 
second av cubic feet or avd pounds of water flow out of the 
orifice, and this amount of water has gained velocity v. To 
impart velocity v to avd pounds in one second requires a force 
equal to adv X v -f- g pounds- weight, according to equation (5), 
and, of course, the jet must push backwards upon the tank with 



HYDRAULICS. 



259 



an equal force. Therefore the reaction of the jet is adv 2 /g 
pounds- weight ; but the velocity of efflux and difference of -pres- 
sure p[ = p' — p" of equation (68)] satisfy the equation 



2g 



dv 2 = p 



so that 



g 



adv 2 = 2pa. 



When a jet of water strikes an obstacle so as to be brought to 
rest, it exerts a force equal to adv 2 /g on the obstacle. This is 
evident when we consider that the force which is exerted by 
a jet as the water is brought to rest is equal to the force which 
must be exerted on the water to set it in motion at the point 
where the jet is produced. If the jet strikes a flat plate so as to 
rebound in a direction at right angles to its original velocity, 
as indicated in Fig. 142, then it exerts the same force as it would 



■^^^w^^ 




Fig. 142. 



Fig. 143. 



exert if it were brought to rest, because it loses all of its velocity 
in the original direction. If the jet strikes a curved plate as 
indicated in Fig. 143 so as to rebound in an opposite direction 
with unchanged velocity (gliding along the curved plate without 
friction), then it would exert twice as much force as it would 
exert if it were brought to rest, because it loses its original veloc- 
ity and gains an equal amount in the opposite direction. 

131. The Pitot tube. A glass tube drawn to a moderately fine 
point is placed in a stream of water moving at velocity v, as 
shown in Fig. 144. In accordance with what is stated above 
concerning the reaction and impact of a jet, the water in the tube 



260 



ELEMENTS OF MECHANICS. 



must stand above the level of the stream at a height h which 
is approximately twice as great as the height which would cause 
an efflux velocity equal to v. That is, the velocity of the stream 
is approximately equal to V gh, according to Art. 127. This 
device is called the Pilot tube, it is frequently used for measuring 
the velocity of streams,* and, when so used, it is usually arranged 





Fig. 144. 



Fig- 145. 



as shown in Fig. 145, so as to bring the difference of level h into 
a convenient position for measurement. The tube A, Fig. 145, 
has its point directed against the stream, and the tube B has its 
point directed at right angles to the stream. By drawing the de- 
vice, Fig. 145, through still water at a known velocity, or by 
using it to measure a velocity which has been determined by 
other means, it has been found that its indications are accurate 
to about one per cent, when the tubes are drawn to small points, 
as shown in Fig. 145. 

132. Gauging of streams. To gauge a stream is to determine 
the volume of water discharged by the stream per second. This 
determination depends upon the measurement of the sectional 
area a of the stream and of the mean velocity v of the stream, 
and the discharge rate of the stream is equal to av according to 
equation (63). 

*Other methods for measuring the velocity of a stream are often used in practice. 
See, for example, Merriman's Hydraulics. 



HYDRAULICS. 26 1 

Small streams are usually gauged by means of an orifice in a 
temporary dam.* Let* x be the distance of the center of the 
orifice beneath the surface of the water in the dam, then the 
velocity of efflux would be equal to V 2gx if the water were f ric- 
tionless, and the product of this velocity and the area of the ori- 
fice a would be the discharge rate if the flow in the orifice were 
lamellar. Experiments show that the mean velocity of a water 
jet flowing from a sharp edged orifice like that shown in Fig. 132 
is about 0.98 of the value {V 2gx) corresponding to ideal friction- 
less flow ; and experiment shows that the cross-sectional area of the 
jet at a short distance from the orifice (where the flow becomes 
approximately lamellar) is about 0.62 of the area of the orifice, 
provided the orifice has sharp edges and is in the middle of a 
flat wall. Therefore the rate of discharge from an orifice like that 
shown in Fig. 132 is approximately equal to 0.98 X 0.62 Y^aV igx. 

A large river is gauged by determining the cross-sectional area 
of the river and measuring the velocity of the water at a large 
number of points in the section so as to determine the average 
velocity. The velocity of the current is sometimes measured by 
means of floats, sometimes by means of Pitot tubes, and some- 
times by means of a so-called current meter which consists of a 
rotating wheel like a screw propeller which drives a speed counting 
device. The current meter has to be calibrated by observing its 
speeds when it is dragged through still water at various known 
velocities. 

133. Fluid friction. The dragging forces which oppose the 
motion of a body through the air or water, and the dragging forces 
which oppose the flow of fluids through pipes and channels are 
due to a type of friction which is called fluid friction. 

Friction of fluids in pipes and channels. There are two fairly 
distinct actions which are involved in the friction of fluids in pipes 
and channels, and, although these two actions always exist to- 

*The arrangement called a weir is a notch in the top of a temporary dam, and the 
formulas for calculating the discharge rate over a weir may be found in any treatise 
on Hydraulics. 



262 



ELEMENTS OF MECHANICS. 



gether, it is instructive to consider two extreme cases in which 
the two actions are approximately separated. 

Viscous Friction. When a fluid flows through a very small, 
smooth-bore pipe, the loss of pressure is proportional to the rate 
of discharge, or to the mean velocity of flow of the fluid in the 
pipe. This fact was first established by Poiseuille (1843). In 
fact, for this case we have 



P 



ttR* 



(70) 



in which I is the length of the tube in feet, R is the radius of its 
bore in fractions of a foot, Q is the volume of liquid in fractions 
of a cubic foot discharged per second, and r\ is a constant called 
the coefficient of viscosity of the liquid. It is evident from this 
equation that the loss of pressure due to viscous friction is very 
small indeed when the radius R of the tube is moderately large. 
In fact, viscous friction is nearly always negligible under practical 
conditions. A full discussion of equation (70) and a definition 
of the coefficient of viscosity are given in Arts. 134 and 135. 

Eddy Friction* Consider a series of chambers, A BCD, Fig. 
146, communicating with each other through narrow orifices, and 




Fig. 146. 

let us suppose water to flow through this series of chambers. As 
the water enters an orifice it gains a certain amount of velocity v, 
and decreases in pressure by the amount i/2g X dv 2 , according 
to Art. 127. The velocity so gained is lost by eddy action in 
the next chamber, and when the water flows through the next 
orifice it must gain velocity anew and suffer a corresponding 

*A fluid entirely devoid of viscosity would not form eddies, so that all fluid fric- 
tion is due to viscosity, directly or indirectly. 



HYDRAULICS. 263 

drop in pressure, as before. It is therefore evident that the drop 
of pressure through the series of chambers, A B CD, is proportional 
to the square of the rate of discharge. This law of eddy friction is 
verified by experiment for a series of chambers as shown in Fig. 
146, where the eddies are definitely localized. In an ordinary 
pipe, however, there is a tendency for the eddy movements to be- 
come finer grained, as it were, with increasing velocity; that is, 
with increased velocity, a given particle of fluid acquires velocity 
and loses it again an increased number of times in traveling a 
given distance. The consequence of this fact is that the loss of 
pressure due to eddy friction increases more rapidly than in pro- 
portion to the square of the rate of discharge. 

In all ordinary cases of the flow of fluids through pipes and 
channels, eddy friction is very much larger than viscous friction, 
and the practical formula for calculating the loss of pressure due 
to the flow of a fluid through a given length of pipe of a given 
size is based upon the assumption that the loss of pressure is pro- 
portional to the density of the fluid and to the square of the rate 
of discharge, or indeed, to the square of the velocity of the fluid 
if the pipe is of uniform size. 

Practical formula for calculating the frictional loss of pressure 
due to the flow of water or gas through a pipe. The formula 
which is used in practice for calculating the frictional loss of 
pressure in a pipe is only approximately true and therefore the 
formula has no rigorous derivation. The only thing to be done 
in connection with it is to exhibit its meaning clearly, which is 
the purpose of the following argument. The flow of a fluid over 
a surface, such as the interior walls of a pipe, is opposed by a force 
which is approximately proportional to the area of the surface, 
to the density of the fluid and to the square of the velocity at 
which the fluid is flowing. Therefore, we may write 

F = kadv 2 (i) 

in which a is the area of the surface in square feet, d is the density 
of the fluid in pounds per cubic foot, v is the velocity of flow in 



264 ELEMENTS OF MECHANICS. 

feet per second, and F is the opposing force in pounds- weight. 
The quantity k is sometimes called the coefficient of friction of 
the moving fluid against the walls of the pipe. It depends greatly 
upon the degree of roughness of the walls. 

Consider a pipe of which the length is L feet and of which the 
inside diameter is D feet. The total area of interior walls of 
this pipe is irDL square feet, so that, using irDL for a in equa- 
tion (i), we have F = kirDLdv 2 for the total opposing force 
acting on a fluid of density d which flows through the pipe at 
velocity v. This opposing force is equal to the difference of 
pressure at the two ends of the pipe multiplied by the sectional 
area of the bore of the pipe. Therefore, using p for the loss of 
pressure due to friction, we have 

ttD 2 
F = p = kirDLdv 2 (ii) 

whence 

P = ± ^~ (70 

in which p is the frictional loss of pressure in pounds per square 
foot in a pipe L feet long and D feet in internal diameter, v is 
the velocity of the fluid in feet per second, and d is the density of 
the fluid in pounds per cubic foot. The value of k is about 
0.000082 for water in ordinary cast-iron pipes and about 0.0000557 
for air in cast-iron pipes. 

Elbows and valves cause excessive eddies and therefore an 
excessive loss of pressure by friction. Methods for estimating the 
effects of elbows and valves are given in standard works on 
hydraulics. 

Example 1. An iron pipe one foot in diameter and 10,000 
feet long discharges 4.25 cubic feet per second of water when the 
pressure at one end is 6,000 pounds per square foot greater than 
the pressure at the other end. A discharge of 4.25 cubic feet 
per second corresponds to a velocity of 5.41 feet per second in 
the pipe (= v). The density of the water is 62 j4 pounds per 



HYDRAULICS. 265 

cubic foot (= d). Substituting these values in equation (71) 
and we find for the coefficient k the value 0.000082. 

Example 2. Compressed air at a mean pressure of 5.42 atmos- 
pheres (density of 0.406 pounds per cubic foot) is forced through 
15,000 feet of pipe 8 inches inside diameter at a velocity of 19.32 
feet per second with a difference in pressure of 5.29 pounds per 
square inch between the two ends of the pipe. Reducing these 
data to the units employed in this chapter and substituting in 
equation (71), we have for the coefficient k the value 0.0000557, 

These two examples indicate the method of determining the 
approximate value of the coefficient k under given conditions. 

134. Definition of the coefficient of viscosity of a fluid. Consider a thin layer 
of fluid of thickness x between two flat plates A A and 55 as shown in Fig. 147 
and suppose that the plate A B is moving at velocity v as indicated by the arrows. 
If the fluid between the plates A B were a viscous liquid like syrup, it is evident 
that a very considerable force would have to be exerted upon the plate A A to 
keep it in motion; in fact any fluid whatever, whether liquid or gas, is more or less 
like syrup in this respect, and the force F with which the motion of the plate is 

A * > -*"■ > A 

MMMBB««aBSMi«MHaflHBBBPp«ll IIIMBHII I ■■P1B__— =£>- V 



Fig. 147. 

opposed by the fluid is proportional to its area a, to its velocity v and inversely pro- 
portional to the distance x between the plates. That is 

F = V f (72) 

in which the proportionality factor 17 is called the coefficient of viscosity of the fluid. 
Examples. The coefficient of viscosity of water at ordinary room temperature is 
0.0000215 and the coefficient of viscosity of good machine oil is about 0.00085; F, a, 
v and x being expressed in terms of the units specified at the beginning of this chap- 
ter. It may seem, therefore, that water would be a better lubricant than the oil, but 
a layer of water would quickly flow out from between a shaft and a bearing surface, 
whereas a rotating shaft continually carries a fresh supply of a viscous liquid like 
oil into the space between the shaft and the bearing surface. 

135. Flow of a viscous liquid through a small smooth-bore tube. Let R be 

the radius of the bore of the tube, I the length of the tube, p the difference of pres- 
sure of the liquid at the ends of the tube, and v the velocity of the liquid at a point 



266 



ELEMENTS OF MECHANICS. 



distant r from the axis of the tube. Consider a cylindrical portion of the moving 
liquid of radius r and coaxial with the tube. The surface of this cylindrical portion of 
liquid moves as a solid rod through the tube at velocity v. Similarly, the cylindrical 
surface of radius r-\-Ar moves through the tube as a hollow shell at velocity v—Av. 
The layer of liquid between this rod and shell is under the same conditions of motion 
as the layer of liquid between the plates A A and BB in Fig. 147. Therefore, 
writing Av for v in equation (72), writing Ar for x, and writing 2irrl for a we have 

2trrl • Av 
* = "— A," 



where F is the force which must act on the end of the rod to overcome the viscous 
drag; but this force is equal to the area of the end of the rod multiplied by p, so that 

2-rrrl - Av 



ir r < 



Ar 



whence 



dv 
dr 

4Vl 



27)1 

+ a constant 



but when r = R, v =0, so that the constant of integration is equal to —pR 2 /4Vl 
and therefore 

pr 2 pR 2 






4.7)1 47)1 



(i) 



The velocity at each part of the tube is thus determined. To find the volume V 
of fluid discharged in time r, consider a section of the tube, Fig. 148. The velocity 

over all the area, 2irrAr, of the dotted annulus, is 
v, so that the volume A V, flowing across this an- 
nulus in time r, is AV = 2TrrAr • v • r. Substitut- 
ing v from (i), we have 




Fig. 148. 



dV 



'P T 3, TtpR 2 T 

r A dr : — rdr 



2l7) 



IT p 
2ll) 



T f n S r s dr 



2l7) 

2ly Jo 



rdr 



Therefore 



V = 



7rpR*T 
Slv 



(ii) 



Problems. 
164. Find the mean velocity at which water must flow in a 
canal 20 feet wide and 6 feet deep, in order that the rate of dis- 
charge may be 500 cubic feet per second. 



HYDRAULICS. 267 

How many acres of storage basin would be required to store 
an amount of water sufficient to maintain this flow of water for 
24 hours, the average depth of the water in the storage basin to 
be 10 feet? Ans. 4.17 feet per second; 99.2 acres. 

165. How much work in foot-pounds is required to pump 
10,000 cubic inches of water into a reservoir in which the pres- 
sure stands at the constant value of 150 pounds per square inch 
above atmospheric pressure? Ans. 125,000 foot-pounds. 

Note. In equation (65) p is to be expressed in pounds per square foot and W 
is expressed in foot-pounds of energy per cubic foot of liquid. 

166. The velocity of a water jet is 200 feet per second, what 
is the kinetic energy of the water in foot-pounds per cubic inch 
Ans. 22.6 foot-pounds per cubic inch. 

Note. In equation (67) the density d is expressed in pounds per cubic foot, the 
velocity v is expressed in feet per second, g is the acceleration of gravity in feet per 
second per second, and W" is the kinetic energy in foot-pounds per cubic foot of 
fluid. 

167. Calculate the velocity of erflux of kerosene from a vessel 
in which the pressure is 52 pounds per square inch above atmos- 
phere pressure. The density of kerosene is 0.03 pound per cubic 
inch. Ans. 96.2 feet per second. 

168. Water flows in a 12-inch main at a velocity of 4 feet per 
second and encounters a partly closed valve through which the 
section of the stream is reduced to 0.36 square foot. Calculate 
the loss of pressure at the valve due to friction. Ans. 0.408 
pounds per square inch. 

Note. As the water enters the narrow passageway in the valve, its velocity in- 
creases by a definite amount, and its pressure falls off accordingly, as explained in 
Arts. 128 and 129. As the water issues from the narrow passageway, it retains its 
velocity as a jet flowing through the surrounding water, so that its pressure does not 
rise again, and the excess of velocity is then destroyed by eddy action. Therefore 
the loss of pressure through the valve is approximately equal to the drop of pressure 
due to the increased velocity of the water as it enters the narrow passage. 

169. A street water-main 7 inches inside diameter has a throat 
3 inches in diameter inserted in it. The flow of water through 
the pipe is 1 x / 2 cubic feet per second and the pressure in the 7-inch 
pipe is 90 pounds per square inch. What is the pressure in the 



268 



ELEMENTS OF MECHANICS. 



throat in pounds per square inch, ignoring friction? Ans. 84 
pounds per square inch. 

170. The difference in level, h, Fig. ijop, is observed to be 
6 inches. Calculate the rate of discharge of water through the 
pipe in cubic feet per second. Ans. 1 cubic foot per second. 

Note. The specific gravity of mercury is 13.6, and the tube AB is entirely 
filled with water above the surface of the mercury. 

171. A pair of Pitot tubes is placed in the stream of air issuing 
from a fan blower, as shown in Fig. ijip, and the difference in 









. air ' ' 




' '• • V 


current 





















Fig. I'jop. 



Fig. 171^. 



level, h, is observed to be 2}£ inches, the tubes being filled with 
water. Calculate the velocity of the air stream. Ans. 72.2 
feet per second. 

Note. In this problem ignore the compressibility of the air, and assume its 
specific gravity to be 0.08 pound per cubic foot. 

172. A temporary dam of thin boards is built across a stream, 
a circular hole one foot in diameter is cut through the dam, and 
the water in the dam rises to a height of 1.5 feet above the center 
of the hole. Calculate the discharge rate of the stream in 
cubic feet per second. Ans. 4.67 cubic feet per second. 

173. A supply of 10 cubic feet per second of water is to be 
brought from a reservior to a center of distribution in a small 



HYDRAULICS. 269 

city. The height of the surface of the reservoir above the point 
of distribution in the city is 350 feet and it is desired to have an 
available head of 150 feet at the center of distribution. Required 
the size of pipe that is necessary to deliver the water, the length 
of pipe being 16,000 feet. Ans. Diameter of pipe required is 
1.336 feet. 

174. A water tank is installed for the protection of a factory 
against fire. The water level in the tank is 75 feet above a cer- 
tain fire hydrant in the building. The pipe leading from the tank 
to the fire hydrant consists of 150 feet of 4-inch pipe in which 
there is one Pratt and Cady check valve, and four short- turn 
ells; and 200 feet of 3 -inch pipe in which there are three short- 
turn ells. Find the number of gallons of water per second that 
can be delivered at the hydrant allowing a loss of head of 25 feet 
in the pipe. Ans. 3.16 gallons per second. 

Note. A four-inch Pratt and Cady check valve has a resistance equivalent to 25 
feet of four-inch pipe, and short turn ells each have a resistance equivalent to 4 feet 
of pipe of the same size. 



PART II. 

THE THEORY OF HEAT. 



CHAPTER I. 

TEMPERATURE. THERMAL EXPANSION. 

1. Thermal equilibrium; temperature. The most important 
single fact in connection with the study of the phenomena of 
heat is that a substance, or a system of substances, settles to a 
quiescent state in which there is no tendency to further change 
of any kind, when it is left to itself and shielded from all outside 
disturbing .influences. This quiescent state is called a state of 
thermal equilibrium. For example, the various objects in a closed 
room settle to thermal equilibrium; when a piece of red-hot iron 
is thrown into a pail of water, the mixture, at first turbulent, 
becomes more and more quiet and finally reaches a state of 
thermal equilibrium. 

A number of bodies which have settled to a common state of 
thermal equilibrium are said to have the same temperature. Thus, 
a number of bodies left together in a closed room have the same 
temperature. 

Although the various objects in a closed room are at the same 
temperature, some of the objects may feel warmer or cooler than 
the others. A piece of warm metal imparts heat to the hand 
more rapidly than a piece of wood at the same temperature, 
and therefore the piece of metal feels warmer than the piece of 
wood. A piece of cool metal takes heat from the hand more 
rapidly than a piece of wood at the same temperature, and there- 
fore the piece of metal feels cooler than the piece of wood. 

2. Atomic theory of heat and thermodynamics.* In nearly 
every branch of physical science there are two more or less dis- 
tinct methods of attack, namely, (a) a method of attack in which 
the effort is made to develop conceptions of the physical processes 

*The term thermodynamics is here used in its proper significance, meaning the 
whole of the theory of heat except those parts which involve the atomic theory. 

19 2 73 



274 THEORY OF HEAT. 

of nature, and (b) a method of attack in which the attempt is 
made to correlate phenomena on the basis of sensible things, 
things that can be seen and measured. In the theory of heat, 
the first method is represented by the application of the atomic 
theory to the study of heat phenomena, and the second method 
is represented by what is called thermodynamics. In the first 
case one tries to imagine the nature of such a process as the melt- 
ing of ice or the burning of coal, and in the second case one is 
content to measure the amount of heat absorbed or given off 
and to study the physical properties of the substances before 
and after the change has taken place. 

The atomic theory. The theory of heat properly includes the 
whole of chemistry, and every student of elementary chemistry 
is familiar with the use of the atomic theory in enabling one to 
form clear ideas of chemical processes. For example, the burning 
of hydrogen is thought of as the joining together of atoms of 
hydrogen and oxygen forming molecules of water vapor. The 
atomic theory is also of considerable use in giving one clear ideas 
of the physical properties of substances. Thus a gas is supposed 
to consist of a great number of particles in violent to-and-fro 
motion, and the gas exerts pressure against the walls of the con- 
taining vessel because of the bombardment of the walls by the 
rapidly moving molecules of the gas. In addition to these two 
highly developed branches of the atomic theory (chemistry and 
the theory of gases), the atomic theory has been applied in a 
more or less vague but very useful way in the study of a great 
variety of heat phenomena as exemplified in the following quota- 
tion from Tyndall's Heat A Mode of Motion* "When a hammer 
strikes a piece of lead, the motion of the hammer appears to be 
entirely lost. Indeed, in the early days it was supposed that 

This book appeared about 1875; sixth edition revised in 1880. It should be 
read by every student who wishes to understand the phenomena of heat in terms 
of molecular motion. To attempt to develop these general ideas in an elementary 
text on heat is out of the question; an elementary text on heat must be devoted 
primarily to thermodynamics. The quotation above is not given in Tyndall'a 
exact words. 



TEMPERATURE. THERMAL EXPANSION. 275 

what we now call the energy of the hammer was destroyed. But 
there is no loss. The motion of the massive hammer is trans- 
formed into molecular motion in the lead, and here our imagina- 
tion must help us. In a solid body, although the force of cohesion 
holds the atoms together, the atoms are supposed, nevertheless, 
to vibrate within certain limits. The greater the amount of 
mechanical action invested in the body by percussion, compres- 
sion, or friction, the greater will be the rapidity and the wider 
the amplitude of the atomic oscillations. 

11 The atoms or molecules thus vibrating, and, as it were, seeking 
wider room, urge each other apart and cause the body of which 
they are the constituents to increase in volume. By the force of 
cohesion, then, the molecules are held together; by the force of 
heat (molecular vibration) they are pushed asunder * ; and the 
relation of these two antagonistic powers determines whether 
the body is a solid, a liquid or a gas. Beginning with a solid 
substance, every added amount of heat pushes the molecules 
more widely apart; but the force of cohesion acts more and 
more feebly as the distance through which it acts is augmented. 
Therefore, as the expansive effect of heat grows strong, its op- 
ponent, cohesion, grows weak until finally the particles are so 
far loosened from each other as to be at liberty; not only to 
vibrate to and fro across a fixed position, but also to roll or glide 
around each other. Cohesion is not yet entirely destroyed,! but 
it is modified so as to permit the particles of the substance to 
glide over each other. This is the liquid condition of matter. 

11 In the interior of a mass of liquid the motion of every molecule 
is limited and controlled by the molecules which surround it. 
But when sufficient heat is imparted to a liquid at a point the 
molecules break the last fetters of cohesion and fly asunder to 
form a bubble of vapor. At the free surface of a liquid it is 

♦These two statements by Tyndall are not always true. Thus when ice changes 
to water contraction takes place and the average distance between the molecules 
decreases. 

fit is a familiar fact that the different parts of a drop of water cling together, or, 
in other words, the force of cohesion is not entirely absent in water. 



2j6 THEORY OF HEAT. 

easy to conceive that some of the vibrating molecules may escape 
from the liquid and wander about through space. Thus freed 
from the influence of cohesion we have matter in the gaseous form." 

Thermodynamics. To understand the essential features of the 
science of thermodynamics, it is necessary to revert to the dis- 
cussion of work and energy. Whenever a substance, or a system 
of substances, gives up energy which it has in store, the substance 
or system of substances always undergoes change. Thus, the fuel 
which supplies the energy to a steam engine and the food which 
supplies the energy to a horse undergo a chemical change; the 
steam which carries the energy of the fuel from the boiler to 
the engine cools off or undergoes a change of temperature when it 
gives up its energy to the engine ; a clock spring changes its shape 
as it gives up its energy in driving a clock; an elevated store 
of water changes its position as it gives its energy to a water wheel; 
the heavy fly-wheel of a steam engine does the work of the engine 
for a few moments after the steam is shut off and the fly-wheel 
changes its velocity as it gives up its energy. 

Not only does a substance undergo a change when it gives up 
energy by doing work, but a substance which receives energy or 
has work done upon it undergoes a change. Thus, when air is 
compressed in a bicycle pump work is done on the air and the 
air becomes warm; when work is done upon a coin in rubbing it 
upon a board, the coin becomes warm; when work is done upon 
a clock-spring in winding it up, the spring changes its shape; 
when work is done in pumping water, the water changes its posi- 
tion to an elevated tank or issues at a high velocity as from a fire 
nozzle. 

In the chapters on Mechanics, the theory of energy was dis- 
cussed in connection with mechanical changes only, thermal and 
chemical changes being carefully ignored. We are now, however, 
to take up the study of thermal and chemical changes and it is 
important at the outset to understand two things as follows: 

(a) Our study is not to be concerned with thermal and chemical 
actions themselves but with their results. The changes themselves 



TEMPERATURE. THERMAL EXPANSION. 277 

are, as a rule, extremely complicated. Thus, the details of behav- 
ior of the coal and air in a furnace are infinitely complicated. The 
important practical thing, however, is the amount of steam that 
can be produced by a pound of coal, and this depends only upon 
(1) the condition of the water from which the steam is made, 
that is, whether the water is hot or cold to start with, (2) the 
condition of the air and of the coal which are to combine in the 
furnace, (3) the pressure and temperature of the steam which is 
to be produced, and (4) the condition of the flue gases as they 
enter the chimney. That is to say, the only things which it is 
necessary to consider are the things which relate to quiescent sub- 
stances. A quiescent substance may be said to be in a standing 
condition or state, and the whole subject of heat (thermodynamics) 
may be said to refer to changes of state, that is, to changes from 
one quiescent condition to another quiescent condition without 
regard to the details of action which leads from one quiescent 
condition to the other. In studying a change of state of a sub- 
stance, variations of temperature, volume and pressure, changes 
of chemical composition, and, above all, the energy given 
to or taken from the substance during the change, are important 
considerations. Beyond these things but little else is involved 
in the study of thermodynamics. 

(b) The other important thing is that in studying thermal and 
chemical changes we have to do with a new kind of energy. The 
gravitational energy of an elevated store of water can be wholly 
converted into mechanical work*, the energy of two electrically 
charged bodies can be wholly converted into mechanical work 
(for example by allowing the charged bodies to move towards 
each other), the kinetic energy of a moving car can be wholly 
converted into mechanical work, and so on. On the other hand, 
the energy of the hot steam which enters a steam engine from a 
boiler cannot be wholly converted into mechanical work. Any store 

*Any energy which is converted into heat because of friction exists in the form 
of mechanical energy or work before it is so converted, and this fact must be kept 
in mind in connection with the statements above given as to the conversion of 
gravitational and electrical energy into mechanical work. 



278 THEORY OF HEAT. 

of energy which can be wholly converted into mechanical work 
may be called mechanical energy. The energy of the hot steam 
which enters a steam engine from a boiler is called heat energy. 
The important difference between mechanical energy and heat 
energy, namely, that one can be wholly converted into mechanical 
work whereas the other cannot, may be clearly understood in 
terms of the atomic theory: Every particle of a moving car 
travels in the same direction and all of the particles work together 
to produce mechanical effect when the car is stopped; the mole- 
cules of hot steam, however, fly to and fro in every direction, 
and no method can be devised whereby the whole of the energy 
of the molecules of hot steam can be used to produce mechanical 
effect. 

The impossibility of converting the whole of the heat energy 
of steam into mechanical work by the steam engine does not, of 
course, refer to the loss of heat by the cooling of the cylinder by 
the surrounding air. It would seem to be possible to reduce the 
temperature (and heat energy) of a gas to zero by allowing the 
gas to expand indefinitely against a piston, but when the pressure 
of the gas is reduced below the pressure of the surrounding 
atmosphere work cannot be obtained from the gas by the 
expansion, on the contrary, work has to be done to produce 
the expansion. In fact this question of the convertibility of the 
heat energy of a gas into mechanical work is bound up inex- 
tricably with the question as to the temperature and pressure of 
the surrounding region, as will become evident in Chapters V 
and VI. 

3. The dissipation of energy.* Preliminary statement of the 
first law of thermodynamics. In the attempt to exclude all 
thermal changes from the purely mechanical discussion of energy f 
we were confronted by the fact that friction (with its accompany- 
ing thermal changes) is always in evidence everywhere. In 
every actual case of motion, the moving bodies are subject to 

*The dissipation of energy is sometimes spoken of as the degradation of energy 
from any form which is wholly available for the doing of mechanical work into heat. 
tSee Mechanics, Art. 55. 



TEMPERATURE. THERMAL EXPANSION. 279 

friction and to collision, their energy is dissipated, and they come 
to rest. This dissipation of energy is always accompanied by 
the generation of heat, and experience shows that the amount 
of heat generated is equivalent to the energy dissipated (first 
law of thermodynamics) . A more complete discussion of the first 
law of thermodynamics is given in Chapter II. The full significance 
of the law is that heat is a form of energy, and that the principle 
of the conservation of energy is applicable not only to mechanical 
changes but to thermal and chemical changes also. Whenever 
mechanical energy disappears, an equivalent amount of heat is 
produced; and whenever heat energy disappears, as in the ex- 
pansion of the steam against the piston of a steam engine, an 
equivalent amount of mechanical energy comes into existence.* 

It is important to understand that the term "dissipation of 
energy" refers to the conversion of mechanical energy into heat 
by friction or collision, f Thus, energy is dissipated in the bearing 
of a rotating shaft, energy is dissipated when a hammer strikes 
a nail, and so on. The atomic theory enables one to form a clear 
idea of the dissipation of energy. Thus the energy of the regular 
motion of a hammer is converted into energy of irregular J molec- 
ular motion when the hammer strikes a nail. 

*One of the most important steps in the establishment of the principle of the 
conservation of energy in its general form, which includes heat energy, was made by 
Count Rumford in his experiments on the generation of heat in the operation of 
boring cannon. The results of these experiments were published in the Philosoph- 
ical Transactions for 1799. The first clear statement of the principle of the con- 
servation of energy in its general form was published in 1842 by Julius Robert 
Mayer. The celebrated experiments of Joule on the heating of water by the dis- 
sipation of work were commenced in 1840. These experiments are described on 
pages 274-278 of Edser's Heat for Advanced Students. The most accurate inves- 
tigation on the heating of water by the dissipation of mechanical energy up to the 
present time is the work of Rowland in 1879. Rowland's experiments, which are 
discussed in Chapter II, are described in detail on pages 278-281 of Edser's Heat 
for Advanced Students, published by Macmillan & Co., London, 1908. 

fMechanical energy is also dissipated in a wire in which an electric current is 
flowing. 

$A substance in thermal equilibrium exhibits no visible motion and therefore 
a state of thermal equilibrium has been called a quiescent state. Very violent 
molecular motion is supposed, however, to exist when a substance is in thermal 



28o THEORY OF HEAT. 

4. Imparting of heat to a substance and the observable effects 
produced thereby. Heat may be imparted to a substance by 
the dissipation of mechanical energy in the substance. Thus, 
heat may be imparted to a coin by rubbing it on a board. Heat 
may also be imparted to a substance by placing it in contact 
with a hotter substance, Thus, heat is imparted to a tea-kettle 
which is placed upon a hot stove. When heat is imparted to a 
substance, the following observable effects may be produced: 

(i) The temperature of the substance may rise. Thus, when 
heat is imparted to a piece of iron or to a vessel of cold water, 
the temperature of the iron or water rises. When, however, heat 
is imparted to ice at it's melting point, some of the ice is converted 
into water but no change of temperature is produced. 

(2) The substance may expand or contract. Most substances 
expand with rise of temperature. There are, however, several 
exceptions to this general rule. Thus, water contracts as its 
temperature rises from the freezing point to about 4°C. at which 
temperature the density of water is a maximum. The expansion 
of a gas with rise of temperature is very much greater than the 
expansion of a liquid or a solid. 

equilibrium but this molecular motion is of the same average character in every 
part of the substance. 

It requires some power of imagination to think of a substance as being composed 
of a great number of small particles (molecules) in incessant and irregular motion, 
and to think of the energy of a moving hammer as still existing by virtue of an 
increased violence of molecular motion after a hammer blow. Every student of 
physics should see the irregular and incessant to-and-fro motion of very fine particles 
suspended in water, using a good microscope. This motion was discovered by the 
English botanist, Brown, in 1827, and it is called the Brolvnian motion. The Brown- 
ian motion is the irregular molecular motion of the water rendered visible (and 
greatly reduced in amplitude) by the small suspended particles. 

To see the Brownian motion, grind a small amount of insoluble carmine in a few 
drops of water by rubbing with the finger in a shallow dish, place a drop of the 
mixture on a microscope slide, and use a magnifying power of about 400 diameters. 
The particles in India ink are much finer than the particles of carmine and a higher 
magnifying power is required to see them. 

A very interesting discussion of the present position of the atomic theory is 
given by Ernest W. Rutherford in his address before the Physics Section of the 
Winnepeg meeting of the British Association for the Advancement of Science. 
This address is published in Science, new series, Vol. 30, pages 289-302, September 
3. 1909. 



TEMPERATURE. THERMAL EXPANSION. 28 1 

(3) The substance may melt or vaporize without change of 
temperature. Thus, when heat is imparted to ice at the melting 
point, part of the ice is converted into water without increasing 
in temperature. When the water in a tea-kettle begins to boil, 
the continued imparting of heat to the tea-kettle from the stove 
converts a portion of the water into steam without increasing 
its temperature. 

(4) The substance may be dissociated. For example, wood 
is converted into charcoal and a smoky gas when it is heated. 
Lime-stone (calcium carbonate) is broken up into quicklime 
(calcium oxide) and carbon dioxide gas when it is heated in a 
lime kiln. 

(5) The subtance, if sufficiently heated, gives off light. 

(6) The substance may exhibit certain electrical phenomena.* 

5. Thermal expansion of gases. Gay Lussac's Law. When a 
number of closed vessels containing different gases all at the same 
pressure are carried from a cool cellar, for example, to a warm 
room, they all suffer the same rise of temperature, and all of the 
gases show the same increase of pressure. That is to say, all gases 
follow the same law of increase of pressure with increase of tem- 
perature, the volumes of the containing vessels being constant. 
This fact was discovered by Gay Lussac and it is called Gay 
Lussac's Law. Following are two precise statements of Gay 
Lussac's Law: 

(a) When equal volumes of various gases are heated under 
constant pressure, they all suffer the same expansion for the 
same rise in temperature. 

(b) When various gases under the same initial pressure are 
heated and not allowed to expand, they all suffer the same in- 
crease of pressure for the same rise in temperature. 

Note. When first discovered, Gay Lussac's Law was thought 
to be exactly true. Very careful measurements, however, show 
perceptible differences of expansion of various gases. 

*See Franklin and MacNutt's Elements of Electricity and Magnetism, appendix C. 



282 THEORY OF HEAT. 

6. The measurement of temperature. To measure a thing is 
to divide it into equal (congruent) parts and to count the parts.* 
One cannot of course divide a force into congruent parts, and 
therefore in a certain fundamental sense forces cannot be mea- 
sured. Neither can one divide a temperature into congruent 
parts, and therefore in a certain fundamental sense temperatures 
cannot be measured. Indeed such things as temperature and 
force can be measured only in terms of their effects. One might, 
for example, take a portion of any gas at the temperature of 
melting ice, and measure the pressure of the gas (without change 
of volume) at the temperature of boiling water, at the tempera- 
ture of melting lead, etc., and any one of these temperatures 
could then be specified numerically by giving the pressure of the 
gas at that temperature, or, better, by giving the ratio of the 
pressure of the gas at that temperature to the pressure of the 
gas at the temperature of melting ice, the volume of the gas 
being unchanged. According to this scheme the ratio of two 
temperatures is the ratio of the pressures of a constant volume of 
gas at the respective temperatures. Thus, if p and p f are the 
pressures of a constant volume of a gas at temperatures T and V 
respectively, then we have by definition 

This provisional definition of temperature ratios will be found 
later to coincide with the thermodynamic definition of tempera- 
ture ratios. 

The air thermometer, f The air thermometer is a device for 
measuring the ratio of two temperatures by observing the pres- 
sures of a constant volume of dry air at the respective tempera- 
tures. The essential parts of the air thermometer are shown in 
Fig. I. The glass or porcelain bulb A contains dry air, and the 
pressure of the air is measured by a syphon barometer BB (or 

*See page 12. 

fThe hydrogen thermometer is the accepted standard. 



TEMPERATURE. THERMAL EXPANSION. 



283 



open tube manometer). The short arm of the barometer at a 

communicates with the bulb A through a tube of fine bore. A 

movable reservoir R containing mercury communicates with the 

barometer through a flexible rubber tube 

and serves to bring the surface of the 

mercury at a to a marked point near the 

end of the fine bore tube, thus keeping 

the volume of the enclosed air sensibly 

constant. 

The bulb A is brought to temperature 
T and the height of the mercury column 
I is measured ; the bulb A is then brought 
to temperature V and the height V of the 
mercury column is again measured. Then, 
according to equation (1), we have 

T'~V 

Standard temperatures. Experiment 
shows that the temperature of pure melt- 
ing ice and the minimum temperature of 
pure steam at a given pressure are in- 
variable. These temperatures, at stand- 
ard atmospheric pressure of 760 millime- 
ters of mercury, are taken as the standard 
temperatures in thermometry. They are called the ice point 
(J) and the steam point (5) respectively. By placing the 
bulb of an air thermometer in melting ice and then in a steam' 
bath (steam at normal atmospheric pressure), the pressures of 
the enclosed air are found to be in the ratio of 1 : 1.367. 
Therefore, according to the above provisional definition of tem- 
perature ratios, we have 

5 

7 = *-3 6 7 (2) 




Fig. 1. 



If the centigrade scale (see Art. 7) is used, the arbitrary value 



284 THEORY OF HEAT. 

100 is assigned to the difference S — I, so that 

S - I = 100 (3) 

From these two equations we find that S = 373 and I = 273 
approximately. The values of S and I are thus known, and any 
other temperature may be determined by measuring its ratio to 
/ (or to S) by means of the air thermometer. Temperatures 
measured in this way are called absolute temperatures. 

Formulation of Gay Lussac's Law and Boyle s Law. When 
temperature is measured by the air thermometer, then the pres- 
sure of a constant volume of any gas is proportional to the abso- 
lute temperature of the gas, or the volume of a gas is proportional 
to the absolute temperature, if the gas is allowed to expand so 
as to keep its pressure constant as the temperature rises. 

According to Boyle's law,* the pressure of a gas is inversely 
proportional to its volume if the temperature is constant, and 
according to Gay Lussac's law, the pressure is directly propor- 
tional to the absolute temperature if the volume is constant. It 
follows from this that the product of pressure and volume is 
proportional to the absolute temperature, that is, we may write 

pv = R r T (4a) 

in which p is the pressure of a gas, v is the volume of the gas, 
T is the absolute temperature of the gas, and R' is a constant 
which depends upon the amount of gas and upon the units in 
terms of which pressure and volume are expressed. A more 
satisfactory form of the above equation is 

pv = MRT (4ZO 

in which M is the mass of the gas in grams and R is a constant 
which is independent of the amount of the gas. 

7. The mercury-in-glass thermometer, j The most convenient 
device for measuring temperature is the ordinary mercury-in- 

*See Mechanics, Art. 101. 

fA good description of the construction of a mercury-in-glass thermometer is 
given on pages 4-14, and special forms of thermometer for indicating maximum 



TEMPERATURE. THERMAL EXPANSION. 285 

glass thermometer with which every one is familiar. A glass 
tube AB, Fig. 2, of fine uniform bore, with a bulb at one end, 
is filled with mercury at a temperature somewhat above the 
steam point and the tube is sealed at A. As the instrument cools, 
the mercury contracts more rapidly than the glass and thus only 
partly fills the stem. The instrument is then placed 
in an ice bath and the position of the surface of the (8\ 
mercury in the stem is marked at /. Then the instru- 
ment is placed in a steam bath at standard atmospheric 
pressure, and the steam point is marked at S. 

In the centigrade scale (Celsius)* which is the scale 
universally used in scientific work, the distance SI is 
divided into 100 equal parts, which divisions are contin- 
ued above 5 and below I. These marks are numbered 
upwards beginning at i" which is number zero. The 
marks below I are numbered negatively from I. 

Any temperature is specified by giving the number of 
the mark at which the mercury stands when the ther- 
mometer is brought to that temperature. For example, 
65 C. (read sixty-five degrees Centigrade) is the tempera- 
ture at which the mercury in a mercury-in-glass ther- 
mometer stands at mark number 65 of the centigrade 
scale. 

Mercury-in-glass temperatures. The indications of an 
accurately constructed mercury-in-glass thermometer are ¥{a 2 
slightly different from air thermometer temperatures 

temperatures and minimum temperatures are described on pages 18-20 of Edser's 
Heat for Advanced Students, Macmillan & Company, London, 1908. 

A device for measuring very high temperatures is called a pyrometer. A good 
discussion of the older methods for measuring high temperatures is given in Bur- 
gess's translation of High Temperature Measurements by Le Chatlier and Boudou- 
ard, John Wiley & Sons, New York, 1904. An excellent discussion of optical 
methods for measuring high temperatures is given by Waidner and Burgess, Bulletin 
of the Bureau of Standards, Vol. I, pages 189-254, February, 1905. 

*The only other thermometer scale of which mention need be made is that of 
Fahrenheit in which the distance SI is divided into 180 equal parts, which divisions 
are continued above 5 and below I. These marks are numbered upwards begin- 
ning with the thirty-second mark below I which is number zero. The marks below 
zero are numbered negatively. 



286 



THEORY OF HEAT. 



(reckoned from ice point) because of the irregularities in the 
expansion of mercury and glass, and temperature values as in- 
dicated by an accurate mercury-in-glass thermometer made of 
a standard variety of glass are called mercury-in-glass tem- 
peratures. 

The following table shows air-thermometer temperatures 
(reckoned from ice point) and mercury-in-glass temperatures 
(Jena normal glass) corresponding to hydrogen-thermometer 
temperatures reckoned from the ice point. All three thermom- 
eters agree, of course, at ice point and at steam point, and the 
differences for the intervening temperatures depend upon irregu- 
larities of expansion. Thus, the difference between the hydrogen 
thermometer temperatures and the air thermometer temperatures 
show that these gases do not both expand in exactly the same 
way with rise of temperature, and another difference between 
the hydrogen and the air thermometers which does not appear 
in the table is that the ratio of steam temperature to ice tempera- 
ture as measured by the hydrogen thermometer is slightly dif- 
ferent from the ratio as measured by the air thermometer. 



TABLE.* 
Comparison of Hydrogen, Air and Mercury-in-Glass Temperatures. 



Hydrogen thermometer 

temperatures (reckoned 

from ice point) 

o.° 

10. 

20. 

30. 
40. 
50. 
6o. 
70. 
8o. 
90. 

100. 



Air thermometer 

temperatures (reckoned 

from ice point) 

o.° 

10.007 
20.008 
30.006 
40.001 
49.996 
59.990 
69.986 
79.987 
89.990 
100. 



Mercury-in- glass 

temperatures (Jena 

Normal Glass) 

o.° 

10.056 
20.091 
30.109 
40. in 
50.103 
60.086 
70.064 
80.041 
90.018 
100. 



Standard thermometers. It is of course impossible to construct 
a thermometer so that the bore of the stem is perfectly uniform, 
and slight errors are always made in the location of the ice and 
steam points and in the marking of the divisions on the stem. 

*From Landolt and Bornstein's Physikalisch- Chemisette Tabellen, page 93. 



TEMPERATURE. THERMAL EXPANSION. 287 

A standard thermometer is a thermometer of which the errors 
have been determined* so that the true mercury-in-glass tempera- 
ture corresponding to any given reading is known. No ther- 
mometer which has not been standardized is to be depended 
upon for work of even moderate accuracy, f 

8. Thermal expansion of liquids and solids. In general, 
liquids and solids expand with rise of temperature. This is 
illustrated by the fact that a long line of steam pipe has to be 
provided with a telescope joint to allow for expansion and con- 
traction inasmuch as the temperature of the pipe is apt to be 
changed at any time from ordinary air temperature to steam 
temperature when the steam is turned on, or from steam tem- 
perature to air temperature when the steam is turned off. The 
movement of the mercury column in the stem of a thermometer 
shows that mercury expands more rapidly than glass as the tem- 
perature rises. The expansion of the glass causes the bulb to 
grow larger but the greater expansion of the mercury causes the 
mercury to rise in the stem. The expansion of a gas (at constant 
pressure) is very much greater than the expansion of a liquid or 
solid, and all gases expand very nearly alike (Gay Lussac's Law), 
whereas every liquid and every solid exhibits characteristic pe- 
culiarities, expanding more rapidly at certain temperatures than 
at others, and in some cases actually contracting with rise of 
temperature. Most liquids exhibit marked irregularities of 
expansion near their freezing points. Thus, water contracts as 
it is heated from o°C. to 4°C. at which temperature the volume 
of a given mass of water is a minimum or its density is a maxi- 
mum; and beyond 4°C. water increases in volume with rise of 
temperature, at first slowly and then more and more rapidly as 
the temperature rises. The ordinates of the curve W in Fig. 3 

*A good discussion of the standardization of a mercury-in-glass thermometer is 
given on pages 23-38 of Edser's Heat for Advanced Students. A discussion of the 
use of a mercury-in-glass thermometer is given on pages 140-143 of Franklin, Craw- 
ford and MacNutt Practical Physics, Vol. I. 

fA well-made thermometer can be sent to the United States Bureau of Standards 
Washington, D. C, where it will be standardized for a small fee. 



288 



THEORY OF HEAT. 



show the volumes at various temperatures of an amount of water 
whose volume at o°C. is equal to unity, and the ordinates of the 
curve M show the volumes at various temperatures of an amount 



uo>4$ 




















































1 






























1 
































/ 












1-035 




















fw 












so 

s 






























































1.030 
































I.025 
























































M, 




















1 












/ 






,1.020 














w 






























/ 


f 


















1. 015 













/ 




M/ 


















In 

4*1 








/ 


f 




















I.OJO 


Wl 








// 






















































1.005 




M, 






























Si— 




'w 








































tern 


pera 


ures 















10 20 ^O 40 



50 fx> 70 8q 
Fig. 3- 



90 JOO JIO S20 130 ll40 ^50 



of mercury whose volume at o°C. is equal to unity. The curve 
M is not a straight line, but its curvature is imperceptible in so 
small a figure. 



TEMPERATURE. THERMAL EXPANSION. 



289 



Coefficient of linear expansion.* The ordinates of the curve 
cc in Fig. 4 represent the lengths of a metal bar at various tem- 
peratures, L being the length at o°C. and L t being the length 
at /°C. The curvature of cc is greatly exaggerated in this 




Fig. 4. 

figure, and for most practical purposes the portion cc of the 
curve may be treated as a straight line, that is to say, the increase 
of length L t — L of the bar from o°C. to t°C. may be considered 
to be proportional to /. This increase in length is also propor- 
tional to the initial length L of the bar because each unit length 
of bar expands by the same amount. Therefore we may write 



or 



L t — L = aLj 
L, = L„(i + at) 



(S) 



The proportionality factor a is called the coefficient of linear ex- 
pansion of the substance, and it is equal to the increase of length of 

♦Coefficients of linear expansion and coefficients of cubic expansion of a great 
variety of substances are given in John Castell-Evans' Physico- Chemical Tables, 
and in Landolt and Bornstein's Physikalisch-Chemische Tabellen. Every student 
of physics and chemistry and every engineer should have access to these tables. 



290 



THEORY OF HEAT. 



a bar of which the initial length is unity when the temperature 
of the bar is raised one degree. 

In calculating the length of a bar from equation (5), greater 
accuracy may be obtained by treating the curve of expansion as 
if it were the straight line aa in Fig. 5 instead of the straight line 




aa in Fig. 4. To do this, the value of a is calculated from equa- 
tion (5) in terms of the observed lengths of the bar at zero and 
at any chosen temperature t\ giving 



a = 



Lj 



(6) 



When so determined, the value of a is called the mean coefficient 
of expansion for the range of temperature from zero to t' ', and when 
this, value of a is used in equation (5) the length of the bar at any 
temperature is represented by the ordinate of the curve aa in 
Fig. 5. 

Some idea of the accuracy with which equation (5) represents the expansion of 
a substance may be obtained from the following example. The mean coefficient 
of linear expansion of a certain sample of annealed steel between o°C. and ioo°C. 
was found by Benoit to be 1.0877X10- 5 per degree centigrade, whereas the coeffi- 
cient of linear expansion of the steel at o°C. [value of a in equation (5) to give the 



TEMPERATURE. THERMAL EXPANSION. 29 1 

dotted line in Fig. 4] was found to be 1.0354X10" 5 . For purposes of very accurate 
calculations, the expansion of this sample of steel may be represented by the formula 

Lt =Lo{i +at + a'l 2 ) (7) 

in which a is equal to 1.0354X10"° and a' is equal to 5.23 Xio -9 . To use this equa- 
tion is to consider the curve of expansion to be a parabola. It must be remembered 
however, that the use of equation (7) gives approximate results, even when it is 
applied to the identical sample of steel which was used in the determination of a 
and a.', especially if the calculation is made for a very high temperature; and of 
course, calculations based upon equations (5) are only approximate. 

Coefficient of cubic expansion. Let V be the volume of a 
given substance at o°C. and V t its volume at t°C The increase 
of volume from o°C. to t°C. is V t — V , and it is accurately 
proportional to the initial volume of the substance and approxi- 
mately proportional to the rise of temperature t\ that is 

v t -v = $V t 

or 

V t = V (i+pt) (8) 

The proportionality factor fi is called the coefficient of cubic ex- 
pansion of the substance. It is equal to the increase of volume of 
a portion of the substance of which the initial volume is unity 
when the temperature of the substance is raised one degree. 

The value of /3 is always determined from equation (8) using the 
observed volume, V , at zero, and the observed volume V/ at 
t'°C, giving 

V< - V 



& 



VJ' 



The value of /3 so determined is the mean coefficient of cubic ex- 
pansion of the substance for the range of temperature from o°C. to 

re. 

Relation between coefficients of linear and cubic expansion. The 
coefficient of cubic expansion of a substance is equal to three 
times its coefficient of linear expansion. Consider a cube of the 
substance of which the length of edge at o°C. is L . At t°C. the 
length of edge is L (i + at). The volume at o°C. is V = L S , 
and the volume at t°C. is V t = L \\ + at)\ or V t = L \i -f 3 at 



292 THEORY OF HEAT. 

+ 3 at 2 + at 3 ). The terms 3 a 2 / 2 and at z are negligible, and there- 
fore writing V for L S , we have 

V t = V (i + 3«) 

Comparing this equation with equation (8), it is evident that /3 
is equal to 3 a. 

Peculiarities of expansion of solids. Solids show irregularities 
of expansion which are in some cases as marked as the irregulari- 
ties of expansion of liquids near their freezing points. These 
irregularities occur at what are called transition temperatures, a 
transition temperature for a given substance being a temperature 
below which the substance is in one crystalline form and above 
which the substance is in another crystalline form. The most 
familiar example of a transition temperature is the so-called tem- 
perature of recalescence of steel.* 

Solids exhibit other peculiarities of expansion which are not 
exhibited by liquid and gases. Thus, many solid substances do 
not expand promptly with rise of temperature or contract prompt- 
ly with fall of temperature, the ultimate change of dimensions 
corresponding to a given change in temperature requiring in 
some cases days or even months before it is established. The 
best known example of this time-lag of expansion is furnished 
by ordinary glass. The mercury column of a mercury-in-glass 
thermometer which has been kept for a long time at room tem- 
perature and which is suddenly brought to steam temperature 
rises at first too high, and as the bulb slowly expands to the 
ultimate size which corresponds to steam temperature the mer- 
cury column slowly drops to its correct position. 

A most interesting substance is the non-expansible nickel-steel 
•alloy which was discovered by Guillaume, a nickel-steel contain- 
ing 36% of nickel, and known as invar. Its coefficient of expan- 
sion is less than one tenth of that of ordinary steel. The increase 
in length of a meter scale made of invar when it is heated from 
o°C. to ioo°C. would be a little less than 0.1 of a millimeter, 

*See Art. 28. 



TEMPERATURE. THERMAL EXPANSION. 293 

whereas a meter scale made of ordinary steel would increase in 
length by about 1.3 millimeters for the same rise of temperature. 
This alloy, invar, is very sluggish in its expansion and contrac- 
tion. When the increase of temperature is small the increase 
of length does not reach its full value for the space of two months. 
Therefore when a bar of invar is subjected to fluctuations of 
temperature which are neither very large nor very long con- 
tinued the change of length of the bar is extremely small and for 
many purposes negligible. 

9. Regnault's method for determining the expansion of water 
and of mercury. The density of any given substance is usually 
determined by weighing equal volumes of that substance and 
of water, thus finding the specific gravity of the substance (num- 
ber of times greater its density is than the density of water at 
the same temperature) so that if the density of the water is 
known at the given temperature, the density of the substance 
may be found. A measuring vessel or graduate, such as is used 
by chemists, is usually standardized by weighing the water or 
mercury required to fill it, whence, if the density of the water 
or mercury is known, the volume of the vessel can be calculated. 
Carefully determined values of the density of water at various 
temperatures and of the density of mercury at various tempera- 
tures are, therefore, of great importance. The densities of water 
and mercury have been accurately determined at a given tem- 
perature by weighing measured volumes of water and mercury; 
and the densities at other temperatures have been determined 
by a method due originally to Regnault. In order to understand 
Regnault's method, it is necessary to establish the relationship 
between the volume of a substance at different temperatures 
and its density at different temperatures. Consider a substance 
of mass m of which the volume at o°C. is V and the volume at 
t°C. is V r The density of the substance at o°C. is 

d G = y (0 



294 THEORY OF HEAT, 

and the density of the substance at fC is 



d f = 



m 



CO 



Dividing equation (i) by equation (ii), member by member, we 
have 

d. v. 



d, V 



Cm) 



from which the volume V t can be calculated when V is known 

and when the ratio d /d t has been determined. 

Regnault's method for determining the ratio d /d t , is as 

follows:* 

Two tubes, A and B, open at top and connected by an air tube C 

at bottom, are filled with the liquid as shown in Fig. 6. The tube 
A is placed in a bath at temperature t°C. 
and the tube B is placed in a bath at tem- 
perature o°C, and the vertical distances l 
and /, are measured. Then 



-J 




d, 



(iv) 



This equation is eivdent when we consider 
that the pressure of the air in C exceeds the 
outside air pressure by the amount l d g or 
by the amount l t d t g, where g is the accelera- 
tion of gravity, as explained in the Chapter 
on Hydrostatics. Therefore 

Kd g = l t d t g 



Fig. 6. 



from which equation (iv) follows at once. 
The following tables give the values of density (grams per 
cubic centimeter) and specific volume (cubic centimeters per 

*A full discussion of this method is given in Edser's Heat for Advanced Students, 
pages 71-80. The discussion here given is merely an outline. 



TEMPERATURE. THERMAL EXPANSION. 



295 



gram) of pure air-free water and of pure mercury at various 
temperatures as determined by Thiessen, Scheel and Marek.* 

Densities and Specific Volumes of Water. 





Grams per 


Cubic centi- 




Grams per 


Cubic centi- 


Temperature. 


cubic centi- 


meters per 


Temperature 


cubic centi- 


meters per 




meter. 


gram. 




meter. 


gram. 


o°C. 


O.999874 


I. OOOI27 


55 


O.9S579 


I. OI442 


5 


O.999992 


1.000008 


60 


0.98331 


1. 01697 


10 


0.999736 


I.OOO265 


65 


.098067 


I.01871 


15 


O.999143 


I.OOO857 


70 


O.97790 


1.02260 


20 


O.998252 


I.OOI75I 


75 


0-97495 


I.02569 


25 


O.997098 


I.OO29H 


80 


O.97191 


I.O2890" 


30 


0.995705 


I. OO4314 


85 


O.96876 


I.03224 


35 


O.994098 


I.OO5936 


90 


O.96550 


1-03574 


40 


0.99233 


I.OO773 


95 


O.96212 


1.03938 


45 


0.99035 


I.OO974 


100 


O.95863 


I-043I5 


50 


O.98813 


I.0I20I 









Densities and Specific Volumes of Mercury. 



Temperature. 


Grams per 


Cubic centi- 


Temperature. 


Grams per 


Cubic centi- 




cubic centimeter. 


meters per gram. 




cubic centimeter. 


meters per gram. 


o°C. 


I3-5956 


O.0735532 


110 


13.3284 


O.0750276 


10 


I3-5709 


O.0736869 


120 


I3.3045 


O.O751624 


20 


I3-5463 


O.0738207 


130 


13.2807 


O.0752974 


30 


13.5218 


O.0739544 


I40 


13.2569 


O.O754325 


40 


13-4974 


O.0740882 


150 


I3.233I 


0.0755679 


50 


13-4731 


O.0742221 


l6o 


13.2094 


O.O757035 


60 


I3-4488 


O.0743561 


I70 


13.1858 


O.O758394 


70 


13.4246 


O.07449OI 


180 


13.1621 


0.0759755 


80 


13.4005 


O.0746243 


190 


I3-I385 


O.076112O 


90 


I3-3764 


O.0747586 


200 


I3-II50 


O.O762486 


100 


I3-3524 


O.0748931 


2IO 


I3-09I5 


O.O763857 



10. Some phenomena dependent upon thermal expansion. 

The winds of the earth constitute the most important group of 
phenomena dependent upon thermal expansion. The radiation 
from the sun penetrates through the upper portions of the atmos- 
phere and reaches the ground where it heats the lower layers 
of the air causing them to expand. After a considerable amount 
of this warm air has accumulated near the ground, it gets started 
upwards at a given point, and a chimney-like effect is developed 
which draws the surrounding warm air into the base of the rising 

*These tables are taken from the more complete tables given in Landolt and 
Bornstein's Physikalisch-Chemische Tabellen. 



2g6 



THEORY OF HEAT. 



column. In the region near the base of the rising column of 
warm air the pressure of the air is low because of the low density 
of the great volume of overlying warm air. The region near the 
base of the rising air column is therefore called a region of "low 
barometer." The wind blows towards such a region of "low 
barometer" from all sides, constituting 
what is called a cyclone.* The movement 
of the air which is here described is inten- 
sified by the formation of water vapor 
near the ground, because water vapor is 
only about half as heavy, volume for vol- 
ume, as dry air. 

The draught of a chimney is due to the 
fact that the hot gases in the chimney are 
lighter than the surrounding atmosphere. 
It is a familiar experience that a chimney 
may not draw at all after it has stood 
idle during the summer months, the tem- 
perature of the air in the chimney may 
then be the same or perhaps even less 
than the temperature of the surrounding 
air. As soon as the chimney becomes 
warm, it produces a draught. 

The circulation of a liquid due to local heating may be shown in 
a striking way by heating a flask of cold water, using a very small 
Bunsen flame, as shown in Fig. 7; a few crystals of magenta 
being placed at the bottom of the flask. The water at the bottom 
dissolves the cyrstals of magenta and becomes colored. This 
colored water expands due to the heat of the Bunsen flame and 
rises, causing a circulation of the water in the flask which is 
indicated by the streamers of colored liquid. 

*See Art. 123 of the chapter on Hydraulics. The word cyclone is used popularly 
for the extremely violent local storms which are properly called tornadoes. The 
cyclone covers many thousands of square miles of country and the air movements 
are widespread and usually of moderate intensity. 




Fig. 7- 



TEMPERATURE. THERMAL EXPANSION. 297 

The circulation of a liquid or gas as above described causes 
the rapid distribution of heat throughout the liquid or gas; heat 
is conveyed from one point to another by the moving fluid, and 
the process is called convection. 

Problems. 

1. Suppose that the average velocity of the molecules of a gas 
at a given temperature is 40,000 centimeters per second. Find 
the increase of the average velocity when 10 joules of mechanical 
energy are dissipated in one gram of the substance ; assume that 
all of the heat energy in the substance is kinetic energy of molec- 
ular motion. Ans. 2426 centimeters per second. 

Note. The average molecular velocity of a gas is of course zero inasmuch as 
velocities occur equally in every direction. What is here referred to is the square-root 
of-the-average-square of the molecular velocity, that is to say, a velocity whose 
square multiplied by the mass of the substance and divided by 2 gives the total 
kinetic energy of molecular motion in ergs, velocity being expressed in centimeters 
per second and mass in grams. 

2. The air in the bulb of an air thermometer has a pressure of 
750 millimeters when the bulb is placed in a steam bath (at 
standard atmospheric pressure) and a pressure of 1203 milli- 
meters when the bulb is placed in a bath of lead at its melting 
point. What is the temperature of melting lead reckoned from 
ice point? Ans. 325°C. 

Note. In solving this problem ignore the slight increase of volume of the air 
thermometer bulb between steam temperature and the temperature of melting 
lead. In the accurate use of the air thermometer the expansion of the bulb must 
be taken account of. 

3. The pressure of the air in an air thermometer bulb at the 
steam point (at standard atmospheric pressure) is 1.367 times 
as great as the pressure of the air in the bulb when it is in an 
ice bath. The difference between ice temperature and steam 
temperature is given the arbitrary value of 180 on the Fahren- 
heit scale. Find the absolute temperature of the freezing point 
in Fahrenheit degrees. Ans. 49i°F. 

4. (a) Reduce to Fahrenheit the following centigrade tem- 
peratures: 45 , 12 , and — 20 . (b) Reduce to centigrade the 



298 THEORY OF HEAT. 

following Fahrenheit temperatures: 21 2°, 72°, 3 2° and — 30 
Ans. U3°F.; 53.6°F.; - 4 °F.; ioo°C; 22.2°C; o°C; -34.4°^ 

5. At what temperature do Fahrenheit and Centigrade ther- 
mometers give the same reading? Ans. — 40 . 

6. The stem of a thermometer has upon it a scale of equal 
parts, and the ice point and the steam point of the thermometer 
are observed to be at a distance of 92.6 of these divisions apart. 
(a) At what point will the mercury stand at a temperature of 
67°C? (b) At what point will the mercury stand at a tempera- 
ture of I20°F.? Ans. (a) 62.04. (b) 45.3. 

7. Suppose a thermometer stem to be divided into 140 equal 
spaces between the ice point and the steam point and suppose 
the marks to be numbered upwards from the tenth division below 
the ice point, making the ice point No. 10. Reduce the following 
readings of this thermometer to centigrade and to Fahrenheit: 
150 , 70°, o°, and -20 . Ans. i07°.iC; 42°.8C; -7°.iC; 
-2i°4C; 224°.qF.; ioo^.iF.; iq°.iF.; -6°.6F. 

8. An amount of gas at I5°C. has a volume of 120 c.c. Find 
its volume at 87°C, the pressure being unchanged. Ans. 150 
cubic centimeters. 

9. A volume of hydrogen at n°C. measures 4 liters. The 
gas is heated until its volume is increased to 5 liters without 
changing the pressure. Find the new temperature. Ans. 82°C. 

10. A flask containing air at 760 millimeters pressure is corked 
at 20°C. Find the pressure of the air in the flask after it has 
stood in a steam bath at gS°C, neglecting the slight increase 
of volume of the flask. Ans. 962 millimeters. 

11. The density of dry air at o°G. and 760 millmeters is 
0.001293 gram per cubic centimeter. What is the volume of 
25 grams of air at 25°C. and at a pressure of 730 millimeters? 
Ans. 21,970 cubic centimeters. 

Note. When the pressure, volume and temperature of a gas all change, calcu- 
lations can be most easily made by using the equation 

py_ p'v' 
T ~T r ' 



TEMPERATURE THERMAL EXPANSION. 299 

This form of equation obviates any consideration of the values of the constants 
M and R in equation (4b). 

12. A quantity of gas is collected over mercury in a eudiometer 
tube. The volume of the gas is observed to be 50 cubic centi- 
meters, its temperature is io°C, the level of the mercury in the 
tube is 10 centimeters above the level of the mercury in the basin, 
and a barometer shows that the atmospheric pressure is 750 
millimeters. Find the volume the gas would occupy at o°C. and 
760 millimeters. Ans. 41.25 cubic centimeters. 

13. The material of a toy balloon weighs 50 grams, and the 
gas in the balloon, consisting partly of carbon dioxide, would 
have at o°C. and 760 millimeters pressure, a density of 0.00132 
gram per cubic centimeter. The volume of the gas in the balloon 
is one cubic meter. Find the temperature of the enclosed gas 
which will barely suffice to buoy up the balloon, the outside air 
being at o°C. and 760 millimeters pressure. Ans. ij°C. 

14. The air in a tall building has an average temperature of 
20°C. and the outside air has a temperature of — io°C. The 
average density of the air inside of the building is that of dry 
air at 20°C. and 760 millimeters pressure, and the average den- 
sity of the air outside is that of dry air at — io°C. and 760 milli- 
meters pressure. The inside and outside pressures are equal at 
a point which is 80 meters above the ground floor. Find their 
difference at the ground floor. Ans. approximately 0.805 milli- 
meters. (The correct result is 0.822 millimeters.) 

Note. The density of the air in the building would not be uniform even if the 
temperature were everywhere the same because the pressure cannot be the same 
from top to bottom. The same is true of the outside air, its density would not be 
uniform even though its temperature were everywhere — io°C. The problem is 
to be solved approximately on the assumption that the density of the outside air 
is uniform and that the density of the inside air is uniform. 

The law of decrease of pressure with increase of altitude in the atmosphere when 
the density is not assumed to be uniform is derived as follows for the case in which 
the temperature is uniform; let p be the pressure of the air at a chosen point (the 
origin of coordinates) and 5 its density. Then the pressure at a point distant Ax 
below the origin is 

p+Ap = p+dg.Ax (i) 

so that 

Ap = 8g- A x (ii) 



3°0 THEORY OF HEAT. 

The density of the air (temperature assumed to be constant) is directly proportional 
to the pressure according to Boyle's Law. Therefore we may write 

d = kp (iii) 

in which k is a proportionality factor. Substituting this value of 5 in equation (ii), 
we have 

&p = kpg.&x (iv) 

whence, using differential notation, we have 

dp 

~=kg-dx (v) 

whence, by integration, we find 

log p = kgx-\-a constant (vi) 

or *" • 

p=Ce"9* (vii) 

that is to say, the pressure in an atmosphere of uniform temperature increases 
according to the exponential function Ce kgx of the distance x below the origin of 
coordinates, where C is the value of the pressure at the origin, k is a constant which 
appears in the expression of Boyle's Law [see equation (iii)], and g is the acceler- 
ation of gravity. 

15. The pressure inside of a chimney at its base is less than 
the outside atmospheric pressure by an amount equivalent to a 
column of water j4 inch in height. The chimney is 125 feet 
high. Find the average temperature of the gases in the chimney 
under the following assumptions: The gas in the chimney would 
have a density of 0.00132 gram per cubic centimeter at o°C. and 
760 millimeters pressure; inside gas is assumed to have every- 
where the density corresponding to its unknown temperature and 
a pressure of 760 millimeters; and the outside air is assumed to 
have a uniform density corresponding to dry air at o°C. and 760 
millimeters pressure. Ans. I02°.5 C. 

16. Plot a curve of which the abscissas represent hydrogen 
thermometer temperatures reckoned from ice point and of which 
the ordinates represent the differences between hydrogen ther- 
mometer temperatures and air thermometer temperatures, (b) 
Plot on the same sheet a curve of which the abscissas are hydrogen 
thermometer temperatures (reckoned from ice point) and of 
which the ordinates are the differences between hydrogen ther- 
mometer temperatures and mercury-in-glass temperatures. 



TEMPERATURE. THERMAL EXPANSION. 30 1 

17. A cheap thermometer is placed in a bath with a standard 
thermometer and simultaneous readings of the two are taken as 
the temperature of the bath is slowly increased giving the fol- 
lowing results: 



Cheap thermometer (Fahrenheit) 
Standard (Fahrenheit) 


20°.0 

2i°-56 


32°.o 

3 2°.8i 


48°.o 
49°-Oi 


6o°.o 

6o°.8o 


Cheap thermometer 
Standard 


72°.0 

72°.50 


88°.o 

88°.6i 


IOO°.0 

ioo°. 49 


n6°.o 
n6°.o6 



Plot a curve of which the abscissas represent the readings of the 
cheap thermometer and of which the ordinates represent the 
true corresponding temperatures. 

Note. It is impracticable to make a standard thermometer so that its readings 
give true temperatures directly; the result of the careful standardization of a high- 
grade thermometer is a table of corrections from which the true temperature cor- 
responding to any reading may be inferred. The readings of the standard ther- 
mometer which are given above are supposed to have been reduced in this way to 
true mercury-in-glass temperatures. 

18. A steel meter scale is 99.981 centimeters long at io°C. and 
100.015 centimeters long at 40°C. At what temperature will 
the scale be exactly one meter long, assuming the expansion from 
io°C. to 40°C. to be proportional to the increase of temperature? 
Ans. 26°.8.C. 

19. A piece of soft wrought iron was found by Andrews to 
have a length of 101.5 centimeters at a temperature of ioo°C. 
and a length of 101.77 centimeters at a temperature of 300°C. 
Find the mean coefficient of linear expansion of the iron between 
ioo°C. and 300°C. Ans. 0.0000133. 

Note. The mean coefficient of linear expansion between two temperatures is 
defined as the difference in length at the two temperatures divided by the length 
at the lower temperature and by the difference of temperature. From this defi- 
nition we have 

Lt>=LAi-\-a(t'-t)} 

The value of a in this equation is slightly different from the value of a in the 
equation 

Lt= Lo(i-\-at) 

but the difference is very small and is entirely negligible when one uses a tabulated 
value of a coefficient of linear expansion for purposes of calculation, unless the 
metal to which the calculation applies is known to be identically the same kind of 
metal as that for which the tabulated value of the coefficient was determined. 



302 THEORY OF HEAT. 

Different samples of commercial iron or steel or copper differ slightly in their ex- 
pansion. 

20. An iron steam-pipe is iooo feet long at o°C. and it ranges 
in temperature from — 20°C. to H5°C. What must be the 
range of motion of an expansion joint to provide for expansion? 
Ans. 1.539 feet. 

Note. The coefficient of linear expansion of wrought iron for the given range 
of temperature is 0.0000114 according to Andrews. 

21. A suryevor's steel tape is correct at o°C. A distance as 
measured by the tape at 22°C. is 500 feet. What is the true 
value of the measured distance, coefficient of linear expansion of 
steel being 0.0000111? Ans. 500.1221 feet. 

Note. The measured distance is 500 times as long as the portion of the tape 
between two adjacent foot-marks at the temperature at which the tape is used. 

22. Ordinary steel rails 30 feet long are laid when the air 
temperature is o.°C. What space must be left between the ends 
of the rails to allow for expansion, the maximum summer tem- 
perature of the rails being 5o°C? The coefficient of expansion 
of rail steel is 0.0000113. Ans. 0.017 $ eet - 

Note. In the laying of steel rails provision is usually made for expansion. The 
rails of a long street car line are however frequently welded into one continuous 
piece of steel; and when such a. rail cools to a low temperature it does not shorten 
but is thrown into a state of tension. 

23. A steel bar of one inch section is stretched by an amount 
equal to 0.000226 of its length when subjected to a tension of 
10,000 pounds. What tension would be required to keep this 
bar unchanged in length when it is cooled from 20°C. to — io°C. 
Ans. 15,000 pounds. 

24. Assuming the highest summer temperature as 45°C. and 
the lowest winter temperature as — I5°C, find the range of ex- 
pansion of one of the 1700- foot spans of the Forth Bridge. The 
bridge is made of steel, the coefficient of linear expansion of 
which is about 0.0000113. Ans. 1.1526 feet. 

25. A brass rod is 100 centimeters long at io°C. and 100. 171 
centimeters long at ioo°C. What is the mean coefficient of 
linear expansion of the brass for the given range of temperature? 
Ans. 0.000019. 



TEMPERATURE. THERMAL EXPANSION. 303 

26. A steel shaft is 20 inches in diameter at 70°F. A steel 
collar is to be shrunk upon this shaft. The collar is to be heated 
to 65 o°F. and have at that temperature an inside diameter of 
20.01 inches, so that it may be easily slipped over the shaft. 
Required the inside diameter to which the collar must be turned 
in the shop, shop temperature being 70°F. The coefficient of 
linear expansion of steel is 0.0000113 per degree Centigrade. 
Ans. 19.937 inches. 

27. A copper plat? has an area of 20 square feet at io°C. 
What is its area at 200°C? The coefficient of linear expansion 
of copper is 0.000017 per degree Centigrade. Ans. 20.129 square 
feet. 

28. A glass bottle is weighed as follows: (a) empty, 24.608 
grams; (b) full of mercury at o°C, 258.723 grams; and (c) full 
of mercury at ioo°C, 255.133 grams. Find the coefficient of 
cubic expansion of the glass of which the bottle is made. Ans. 
0.000027. 

Note. The space inside of a vessel increases exactly as if it were a solid piece 
of the material of which the vessel is made. Therefore the mean coefficient of 
cubic expansion of the glass of which the bottle is made is equal to the difference 
in volume of the mercury in the bottle at o°C. and ioo°C. divided by the volume 
of the mercury at zero and by the difference of temperature. 



CHAPTER II. 



CALORIMETRY. 

11. Complete statement of the first law of thermodynamics. 

A given substance is heated by the dissipation of work and brought 
back to its initial state by being cooled by contact with another 
(cooler) substance B. Then, if loss of heat to surrounding bodies 

is carefully avoided, the ther- 
mal effect produced in sub- 
stance B is exactly the same as 
would be produced in it if it had 
been heated directly by the 
dissipation of the original 
amount of work. Therefore, 
a substance which is heated by 
the dissipation of work stores 
something which is equivalent to 
the work and which is called 
heat. The conception of heat 
as the energy of molecular mo- 
tion is explained in Art. 2. 
Mechanical energy can be con- 
verted completely into heat but 
the conversion of heat into me- 
chanical energy is subject to 
important limitations, as ex- 
plained in Chapter V. 

12. The heating of water by the dissipation of mechanical 
energy. The relation between the amount of work dissipated 
in heating water and the rise of temperature produced has been 
determined with great care. The results of Rowland's deter- 
mination are given in the accompanying table and are shown 

304 



110 


1 1 1 

JOULES TO HEAT ONE 










GR/ 


M OF ' 


VATEf! 












120 






































































80 


































60 


































40 


































20 






































TEMPE 


RATUt 


iE RISE 


(hyd 


^OGEN 


SCALE 


\ 



10' J5 3 20 J 25 g 

Fig. 8. 



CALORIMETRY. 



305 



graphically in Fig. 8. The ordinates of the curve in Fig. 8 
represent the amount of work in joules required to raise the tem- 
perature of one gram of water from o°C. to t°C (hydrogen scale). 
This same quantity of work is given in the table in the column 
headed E. 





















1 






















En 


3rgy required to 'raise 'the te 
of one gram of wa"ter|one c 


mperature 
egree 




















at ve 


rious 


temp 


eratu 


•es 






-4.20-J 


oules- 






















































-4.19J 


























































■4.18-d 



























































































JO 15 20 25 30 

TEMPERATURE (.Hydrogen Scale) 

Fig. 9. 



35 



TABLE.* 
Rowland's Determination of the Work Required to Heat Water. 



Temperature rise from o°C. to 


Energy in joules to heat one 
gram of water. 


(0 


(S) 


5° 


21.040 


10° 


42.041 


15° 


63.005 


20° 


83-935 


25° 


104.834 


30° 


125.708 


35° 


146.745 



The ordinates of the curve in Fig. 9 show Rowland's results 
in a slightly different form. For most practical purposes it is 

*From Rowland's results by W. S. Day to the hydrogen scale. (Physical Review, 
Vol. VIII, April, 1898). 



306 THEORY OF HEAT. 

sufficiently exact to take 4.2 joules as the energy required to 
raise the temperature of one gram of water one degree centigrade 
(or 778 foot-pounds to raise the temperature of one pound of 
water one degree Fahrenheit). 

Rowland's determinations were made by driving a rotatory 
paddle about a vertical axis at an observed speed in a vessel of 
water itself mounted on a vertical axis and prevented from turn- 
ing by a cord passing over a pulley to a weight. The torque 
exerted by the paddle is thus equal to the product of the pull of 
the cord (which is equal to the attached weight) and the lever 
arm thereof. This torque multiplied by the angular velocity of 
the paddle in radians per second and by the time gives the work 
expended in heating the water. An accurate thermometer pro- 
jecting through the hollow axle of the paddle indicates the rise 
of temperature of the water.* 

13. Measurement of heat. An amount of heat, for example, 
the amount required to melt a gram of ice, or to raise the tem- 
perature of a gram of lead i°C, is measured when the amount 
of work required to produce the effect has been determined. This 
measurement may be made by the direct determination of the 
work required to produce the given effect. The accomplishment 
of this method of heat measurement is, however, very tedious 
and subject to a very considerable error, f This is partly due 
to the difficulty of measuring work mechanically and partly due 

*Using the data of Count Rumford's experiments (1798), one would find by- 
calculation that about 847 foot-pounds are required to raise the temperature of 
one pound of water one degree. The first accurate and carefully planned deter- 
mination was made by Joule, beginning in 1840. Joule's result was 772 foot-pounds 
per British thermal unit. 

Rowland's experiments, which were carried out in 1879, are described in the 
Proceedings of the American Academy of Arts and Sciences, new series, Vol. 7. A 
good description of this work of Rowland's and of the work of other experimenters 
in the same field, is given on pages 267-286 of Edser's Heat for Advanced Students. 

fThe work spent in any portion of an electric circuit, can be measured with con- 
siderable accuracy and it can be easily applied to the accomplishment of any given 
thermal effect, and this electrical method for measuring heat values in energy units 
is perhaps the most accurate method at present available. 



CALORIMETRY. 



307 



to the difficulty of applying mechanical work wholly to the heat- 
ing of a given substance. 

Practical method of heat measurement. The water calorimeter* 
The water calorimeter is a vessel containing a weighed quantity 
of water W arranged to absorb an amount of heat to be measured. 
Thus if the heat generated by the burning of a weighed quantity 
of coal is to be measured, the water calorimeter is arranged so 
that the whole of the heat generated by the burning of the coal 
may be absorbed by the water of the calorimeter. Figure 10 
shows the water calorimeter as arranged for measuring the 
amount of heat given off by the 
cooling of a weighed amount of 
hot metal or other substance B. 
In this case the hot substance (at 
known initial temperature t) is 
plunged into the water of the 
calorimeter (at known initial tem- 
perature t'), the water is stirred 
vigorously by means of the stirrer 
55, and the final temperature t" 
of the water is observed. The 
quantity of heat which is ab- 
sorbed by the water of the calori- 
meter is calculated as follows: 

(a) Accurate calculation. Let E' be the energy in joules re- 
quired to heat one gram of water from zero to t', and let E" be 
the amount of energy in joules required to heat one gram of 
water from zero to t" (see table on page 305). Then W(E" — E r ) 
is the amount of energy in joules required to heat the whole quan- 
tity of water in the calorimeter from t' to t" , where W is the mass 
of the water in grams. Therefore W{E" — E') is the energy 
value of the heat which has been imparted to the water of the 
calorimeter. It is, of course, necessary to make allowance for 

*Other forms of calorimeter, such as the ice calorimeter and a variety of forms 
of the water calorimeter for special purposes are described on pages 11 7-163 of 
Edser's Heat for Advanced Students. 



81^ 



s-—=-- S 



m 



Fig. io. 



308 THEORY OF HEAT. 

the amount of heat required to raise the temperature of the con- 
taining vessel of the calorimeter. This matter is explained below. 

(b) Approximate calculation. The curve in Fig. 8 is plotted to 
represent accurately the tabulated values on page 305 , and this 
curve is so nearly straight that it is impossible to represent it 
otherwise than by a straight line in so small a diagram as Fig. 8. 
That is to say, the energy required to heat a gram of water is 
approximately proportional to the rise of temperature, and for 
most practical purposes the heat absorbed by a water calorimeter can 
be calculated by multiplying the weight of water in the calorimeter 
by the observed rise of temperature. This gives the amount of heat 
in terms of what may be called the water-unit of heat, namely, 
the amount of heat required to raise the temperature of one gram 
of water one centigrade degree (the calorie), or the amount of 
heat required to raise the temperature of one pound of water one 
Fahrenheit degree (the British thermal unit). 

The amount of heat required to raise the temperature of one 
gram of water one centigrade degree varies slightly with, tempera- 
ture, or in other words the curve in Fig. 8 is not a straight line 
The extent of this variation is shown in Fig. 9 which is derived 
from the results of Rowland's work on the heating of water as 
given in the table on page 305. 

Calorimetric results may be expressed in energy-units or in water- 
units. It is desirable that all results of accurate calorimetric 
work be expressed in energy-units. It is usual, however, to ex- 
press accurately measured heat values in terms of what is called 
the standard calorie, which is the amount of heat required to 
raise the temperature of one gram of water from I4°.5C to I5°.5C. 
The standard calorie is equal to 4.189 joules. See Fig. 9. 

The mechanical equivalent of heat, so called. In the early days 
of the nineteenth century heat was generally thought of as a 
peculiar kind of fluid, and the water-unit of heat was looked upon 
as a truly fundamental thing. So widespread was this point of 
view that Joule's result (1 British thermal unit = 778 foot-pounds) 
was for many years spoken of as the "mechanical equivalent of 



CALORIMETRY. 309 

heat" ; the simple fact is, however, that by the dissipation* of 778 
foot-pounds of work the temperature of one pound of water may 
be raised one Fahrenheit degree. 

Sources of error in the use of the water calorimeter, (a) A 
portion of the heat imparted to the calorimeter is used to raise the 
temperature of the containing vessel, the stirrer and the ther- 
mometer bulb. The vessel and stirrer, being of the same ma- 
terial, are equivalent f to an amount km of water, where k is the 
specific heat (see Art. 14) of the material and m is the combined 
mass of vessel and stirrer. The water equivalent of the thermo- 
meter bulb is calculated in the same way, and in calculating 
the amount of heat absorbed by the calorimeter, the value 
W + km + k'm' is used instead of W as the mass of the water 
in the calorimeter, km being the water equivalent of the vessel 
and stirrer and k'm' being the water equivalent of the thermo- 
meter bulb. 

(b) When the calorimeter is cooler than room temperature it 
absorbs heat from its surroundings and vice versa. This source 
of error is to a great extent obviated by arranging that the initial 
temperature of the calorimeter may be as much below room tem- 
perature as the final temperature of the calorimeter is above 
room temperature, and by making the duration of the experiment 
as short as possible. 

The vessel of the calorimeter CC, Fig. 10, should be made of 
thin polished metal and it should be surrounded by a polished 
metal jacket J J with an air space between to reduce to a mini- 
mum the exchange of heat between the calorimeter and its sur- 
roundings. The calorimeter-vessel is usually supported on 
small wedges of cork. 

(c) In a vessel of water which is being heated there are always 
large local differences of temperature. In order that the indica- 

*See definition of the term dissipation of energy on page 279. 

fAn amount of metal and an amount of water are equivalent to each other in 
the sense here intended when a given amount of heat will produce the same rise 
of temperature of the one or of the other. 



3IO THEORY OF HEAT. 

tions of the thermometer may be accurate, brisk stirring is nec- 
essary. 

14. Specific heat-capacity of a substance.* The number of 
thermal units required to raise the temperature of one gram of a 
substance one centigrade degree is called the specific heat-capacity 
or the specific heat of the substance. Specific heats are ordinarily 
specified in calories per gram of substance per degree rise of tem- 
perature. It is sometimes desirable, however, to express specific 
heats in joules per gram per degree rise of temperature. Thus, 
the specific heat of water at 5°C. is about 4.204 joules per gram 
per degree centigrade (see Fig. 9) . 

The specific heats of most substances vary considerably with 
temperature. Thus, the ordinates of the curve in Fig. 9 show 
the specific heat of water at various temperatures expressed in 
joules per gram per degree centigrade. The mean specific heat 
of a substance for a given range of temperature f to t is the 
amount of heat required to heat one gram of the substance from 
f to / divided by (t — t") , therefore the amount of heat required 
to heat 5 grams of the substance from f to t is equal to kS{t — t") , 
where k is the mean specific heat of the substance, S is its mass 
in grams and (t — f) is the change of temperature. The mean 
specific heat of a substance is usually determined by heating 
S grams of the substance to a known temperature t and plung- 
ing it into a water calorimeter at known temperature t', the 
final temperature f of the mixture being observed. The amount 
of heat given off by the substance in cooling from t to f is 
kS(t — f), and this amount of heat, as measured by the rise of 
temperature of the water of the calorimeter, is equal to W(f — t r ) 
as explained in Art. 13. Placing these two amounts of heat 
equal to each other, we have 

k - -s^^y (*°) 

*The specific heats of a great variety of substances are given in John Castell- 
Evans' Physico- Chemical Tables, and in Landolt and Bornstein's Physikalisch- 
Chemische Tabellen. 



CALORIMETRY. 



311 



in which W is the mass of water in the calorimeter (including the 
water equivalents of vessel and stirrer), 5 is the mass of the 
substance, t is the initial temperature of the substance, t' is the 
initial temperature of water in the calorimeter, and f is the final 
temperature of the mixture. 

The extent to which the specific heat of a substance varies 
with temperature is further exemplified by Fig. 11. The ordi- 
nates of the curves in this figure show the specific heats of iron 
and the specific heats of crystalline carbon at various tempera- 



























c 








0.4 
















^^ 






D 


















,0 


$y 




















0.3 










f 








: 


r 












(- 
< 
Ijj 






4 
























0.2 


X 

to 




a £ 


9 








*y 




















Vi 
*/ 






1 


4%/ 


















0.1 












8£^ 
































































TEM 


3 ERA 


ruRE 


S 













200' 



,400° 60Q a 

Fig. 11. 



$xr 



tures. As a rule the specific heat of a substance increases with 
rise of temperature. 

15. Heat of combustion.* The number of thermal units of 
heat developed by the burning of unit mass of a fuel is called 
the heat of combustion of the fuel. For example, the burning of 
one gram of soft-wood charcoal generates 7070 calories. 

Chemical action is in general accompanied by the development 
or the extinction of heat. Those chemical reactions during the 

*Heats of combustion of a great variety of substances are given in Landolt and 
Bornstein's Physikalisch-Chemicische Tabellen and in John Castell-E vans' Physico- 
Chetnical Tables. 



312 THEORY OF HEAT. 

progress of which heat is generated are called exothermic reactions. 
Those chemical reactions during the progress of which heat is 
absorbed or extinguished are called endothermic reactions. Com- 
bustion is the most familiar example of exothermic reaction. 
A very interesting feature of exothermic reactions is that such a 
reaction does not take place at an extremely high temperature. 
Thus hydrogen and oxygen exist side by side at extremely high 
temperatures without combining to form water vapor; on the 
other hand an endothermic reaction takes place eagerly (as it 
were) at extremely high temperatures. Thus at the tempera- 
ture of the electric arc oxygen and nitrogen combine with ab- 
sorption of heat. If we lived in an extremely high-temperature 
world we could cool our houses by the combustion of nitrogen 
and oxygen. 

Problems. 

29. (a) How many watts are required to raise the temperature 
of 10 liters of water from o°C. to io°C. in 20 minutes? (b) 
How many horse-power are required to raise the temperature of 
10 gallons of water from 32°F. to 6o°F. in 20 minutes? Ans. 
(a) 3 50 watts, (b) 2.76 horse-power. 

Note. Solve this problem approximately by assuming that it takes 4.2 joules 
to raise the temperature of one gram of water i°C. or 778 foot-pounds to raise the 
temperature of one pound of water i°F. Solve the first part of the problem accur- 
ately by the use of the table on page 305. 

30. Niagara Falls is 165 feet high. Calculate the rise of tem- 
perature of the water due to the energy of the fall on the assump- 
tion that the cooling effect of vaporization is zero. Ans. o.2i2°F. 

31. A copper calorimeter weighing 50 grams contains 500 grams 
of water at i6°C. A piece of copper weighing 65 grams is heated 
to ioo°C. and plunged into the water of the calorimeter. The 
resulting temperature is I7°C. Find the specific heat of copper. 
Ans. 0.0935. 

32. A piece of lead weighing 15 pounds at ioo°C. is placed in 
a vessel containing 4 pounds of water at I5°C, and the tempera- 
ture of the mixture finally settles to 24°.oiC. The vessel is made 



CALORIMETRY. 313 

of copper and it weighs j£ pound. What is the specific heat of 
lead, the specific heat of copper being 0.093? Ans. 0.032. 

33. A copper vessel weighing 2 kilograms contains 24 kilo- 
grams of water at 20°C. Into this vessel are dropped at the 
same instant, 2 kilograms of copper at ioo°C, 2.4 kilograms of 
zinc at 6o°C, and 6.4 kilograms of lead at 50°C. Find the re- 
sultant temperature. The specific heat of zinc is 0.093. Ans. 

2I°.2C. 

Note. When there is no question as to the freezing of a portion of the water 
or the boiling of a portion of the water, the simplest argument of a problem of this 
kind is as follows: Let t be the resultant temperature and for the sake of argument 
let us think of t as being higher than any of the given temperatures. Then the 
specific heat of any one of the given substances multiplied by its mass and multiplied 
by {t minus initial temperature of substance) is the amount of heat required to 
raise the substance up to the resultant temperature. Adding all such products 
together gives the total heat required to raise the mixture up to its resultant 
temperature and this total heat is equal to zero. 

34. A piece of steel weighing 6000 pounds is heated to a tem- 
perature of i5oo°F. and plunged into an oil bath at 6o°F. As- 
suming that all of the heat which is given off by the steel in 
being cooled to the final temperature of the bath is absorbed by 
the bath (no portion of it lost in vaporizing the oil), calculate 
the number of pounds of oil required in order that the final 
temperature of the bath may be ioo°F. The specific heat of 
the oil is 0.7 and the mean specific heat of the iron between the 
specified temperatures is 0.2. Ans. 60,000 pounds. 

35. A piece of platinum weighing 100 grams is drawn from a 
furnace and dropped into a water calorimeter at 20°C. and the 
resultant temperature is 59.2°C. The calorimeter vessel is of 
silver and weighs 15 grams (specific heat of silver 0.056) and the 
calorimeter contains 100 grams of water. The mean specific 
heat of platinum between ordinary room temperature and 1100 
or i200°C. is 0.0382. Find the temperature of the furnace. 
Ans. i093°C. 

36. The heat of combustion of hydrogen is 34,000 calories per 
gram. How many calories does this represent per gram of 
oxygen and how many calories per gram of water produced? 



314 THEORY OF HEAT. 

Ans. 4250 calories per gram of oxygen; 3777 calories per gram of 
water. 

37. The heat of combustion of pure charcoal is 4000 British 
thermal units per pound when the product of the combustion is 
carbon monoxide CO and 14,500 British thermal units per pound 
when the product of the combustion is carbon dioxide C0 2 . 
What is the heat of combustion of carbon monoxide per pound? 
Ans. 4500 British thermal units per pound. 

38. Two volumes of carbon monoxide (CO) are mixed with 5 
volumes of air in an explosion chamber, the temperature of the 
mixture being 20°C. and its pressure one atmosphere (14.7 
pounds per square inch). Find the temperature and pressure 
immediately after the mixture is exploded by a spark. The heat 
of combustion of carbon monoxide is 2400 calories per gram of 
CO, the mean specific heat of carbon dioxide (C0 2 ) between the 
initial temperature and the temperature immediately after the 
explosion is 0.25 calories per gram and the mean specific heat of 
nitrogen for the same range of temperature is 0.3 calories per 
gram. Ans. 2428°C; 116 pounds per square inch. 

Note. The specific heats above given are specific heats at constant volume. 

Two volumes of carbon monoxide combine with one volume of oxygen to give 
two volumes of carbon dioxide so that 5 volumes of air contain the proper amount 
of oxygen to convert two volumes of carbon monoxide into carbon dioxide. The 
change of volume due to chemical combination is such that the 7 volumes of gas 
before the explosion would be reduced to 6 volumes after the explosion if the tem- 
perature and pressure were unchanged. 



CHAPTER III. 



THERMAL PROPERTIES OF SOLIDS, LIQUIDS AND GASES. 

16. Melting points and boiling points.* When heat is imparted 
to a solid the temperature rises until the solid begins to melt; the 
temperature then remains constant until all of the substance is 
changed to liquid; the temperature then begins to rise again 
and continues to rise until the liquid boils; the temperature then 
remains constant until the liquid is entirely changed to vapor 
(pressure being unchanged) ; and then the temperature begins to 
temperature 



212* 
32° 



water and steam 




ice and 
waters 



water 



amount of heat imparted to the water 



Fig. 12. 



rise again as heat is continually imparted to the substance. 
There are thus two periods during which heat is imparted to a 
substance without producing rise of temperature, namely, when 
the substance is melting and when the substance is boiling under 
constant pressure. 

Examples. Heat is imparted to very cold ice and the tempera- 
ture rises until the ice begins to melt at o°C, after the ice is all 

♦Extensive tables of melting points and boiling points are given in Castell-Evans' 
Physico- Chemical Tables, and in Landolt and Bornstein Physikalisch-Chemische 
Tabellen. 

315 



316 



THEORY OF HEAT. 



melted the temperature rises to ioo°C, and after the water is 
all converted into steam the temperature of the steam is raised 
by further addition of heat, as shown by the ordinates to the 
curve in Fig. 12. The ordinates in Fig. 13 represent observed 
temperatures of a cooling crucible containing melted zinc. The 
temperature continues to fall until the zinc begins to freeze (at 
41 8°), the temperature then remains constant until all of the 
zinc is frozen after which the temperature again falls. 
486 



470 



460 



450 



440 
a 



! 

•to 4IO 
400 



•390 



3BO 



25 30 35 40 45 5° 55 6g 
seconds 
Fig. 13. 

The melting point of a substance is the temperature at which the 
solid and liquid forms of the substance can exist together in 
thermal equilibrium. This temperature varies but slightly with 
pressure. All substances which like ice expand upon freezing have 
their melting points lowered by pressure, and all substances which 
like wax contract on freezing have their melting points raised 
by pressure. 



SOLIDS, LIQUIDS AND GASES. 



317 



The boilir.g point of a liquid at a given pressure is the tempera- 
ture at which the liquid and its vapor can exist together in equil- 
ibrium. This matter is discussed more fully in the next article. 

Every substance as far as known has a definite boiling point 
at any given pressure; but crystalline substances, only, have 
definite melting points. Thus, ice crystals and water exist side 
by side in equilibrium at a given temperature, whereas amorphous 
substances, such as glass and resin, have no 
definite melting point. These substances 
grow softer and softer with rise of temper- 
ature. 



carbon dioxide 
at constant 
temperature t 



tll>>UUljr<X , 



t 

17. Maximum pressure of a vapor at a 
given temperature and minimum temper- 
ature of a vapor at a given pressure. Con- 
sider a cylinder CC, Fig. 14a, filled with 
carbon dioxide and kept at a fixed tempera- 
ture t while the piston PP is pushed down- Flg- I4a - 
wards. At first the pressure of the gas increases as its volume 
decreases (in accordance with Boyle's Law), but when the pressure 
reaches a certain value (which depends upon the temperature /) 
then continued decrease of volume does not cause further increase 
of pressure but results in the condensation of a portion of the car- 
bon dioxide into a liquid.* There is thus a certain maximum pres- 
sure that gaseous carbon dioxide is capable of exerting at a given tem- 
perature and if an attempt is made to increase its pressure beyond 
this value by compression, a portion of the carbon dioxide condenses 
into liquid form, the remainder being at the same pressure as before. 
The successive changes here described are represented graphically 
in Fig. 146, in which abscissas represent volumes and ordinates 
represent pressures, everything being at a given temperature. 
The substance is wholly gaseous for the portion a of the curve, 
partly gaseous and partly liquid for the portion b, and wholly 
liquid for the portion c. 

*If the temperature does not exceed a certain value which is called the 
critical temperature of ihz substance. See Art. 22. 



3i8 



THEORY OF HEAT. 



The facts which are stated above and which are represented in 
Fig. 14b maybe stated in a slightly different manner as follows; 
A cylinder CC, Fig. 15a is filled with carbon dioxide, and the 



axis of pressures 



!— maximum 
vapor pressure" 
at given temperature' 



axis or 



QQkmw 



Fig. 14&. 

temperature of the whole is slowly decreased, the pressure of the 
carbon dioxide being kept at a constant value by pushing the piston 
downwards if necessary. Under these conditions, the carbon 
dioxide remains wholly in a gaseous or vapor form until a certain 
temperature is reached below which the 
carbon dioxide cannot exist as a gas or 
vapor at the given pressure. When this 
temperature is reached, a continued ab- 
straction of heat causes a portion of the 
carbon dioxide to condense to a liquid,* 
without decreasing its temperature, pres- 
sure being kept at a constant value. 
When, however, all of the carbon dioxide 
is condensed to liquid form, further ab- 
straction of heat causes a further drop 
The successive changes here described are 



carbon dioxide 
at constant 
pressure p 



Fig. 15a. 

of temperature 



*If the given pressure does not exceed what is called the critical pressure of the 
substance. See Art. 22. 



SOLIDS, LIQUIDS AND GASES. 



319 



represented graphically in Fig. 15&. For the portion a of the 
curve the carbon dioxide is wholly in the gaseous form. For 
the portion b it is partly gaseous and partly liquid, and for 
the portion c it is wholly liquid. 

In discussing the change of a substance from a liquid to a gas 
or from a gas to a liquid it is customary to speak of the gaseous 
form of the substance as vapor. When the vapor is at its maxi- 



axis of temperatures 



^minimum vapor 
temperature at 
given pressure 



axis of 



volumes 



Fig. 15&. 

mum pressure for a given temperature or at its minimum tem- 
perature for a given pressure, that is to say, when the pressure 
and temperature of the vapor are such that it would be in equil- 
ibrium with the liquid form of the substance if the liquid form 
were present, then the vapor is said to be a saturated vapor. A 
saturated vapor cannot be cooled without a portion of it being 
condensed if the pressure remains the same. A saturated vapor 
cannot be compressed without a portion of it being condensed 
if the temperature remains the same. 

What is said here of carbon dioxide is true so far as known of 
every substance. Thus, water vapor at a given temperature 
cannot exert more than a certain maximum pressure, or at a 
given pressure it cannot be cooled below a certain minimum tern- 



320 



THEORY OF HEAT. 



perature without condensation. The following tables give the 
maximum pressures and minimum temperatures of water vapor, 
and of anhydrous ammonia.* 

TABLE. 

Pressures and Temperatures of Saturated Water Vapor. 
(Boiling Points of Water at Various Pressures). 



Temp 


Pressure in 


Temp 


Pressure in 


Temp. 


Pressure in 


Temp 


Pressure in 




centimeters. 




centimeters. 




centimeters. 




centimeters. 


-I0°C. 


0.2151 cm. 


50°C. 


9.1978 cm. 


iio°C. 


107.537 cm. 


i70°C. 


596.166 cm. 


0° 


0.4569 


6o° 


14.8885 


120° 


149.128 


180 


754.692 


10° 


0.6971 


. 70 


23.3308 


130° 


203.028 


190 


944.270 


20° 


1-7303 


8o° 


35-4873 


1 40 


271.763- 


200° 


1168.896 


30° 


3-i5io 


90 


52.5468 


1 50 


358.123 


210° 


1432.480 


40° 


5-4865 


100° 


76.0000 


160 


465.162 


230 


2092.640 



TABLE. 

Pressures and Temperatures of Saturated Ammonia Vapor (N H^). 
(Boiling Points of Liquid Ammonia at Various Pressures). 



Temperature 


Pressure in atmospheres. 


Temperature 


Pressure in atmospheres. 


-30° C. 


1. 14 atm. 


20° C. 


8.41 atm. 


— 20° 


1.83 


40° 


15.26 


-10° 


2.82 


6o° 


25-63 


0° 


4.19 


8o° 


40-59 


10° 


6.02 


IOO° 


61.32 



The phenomenon of boiling, f According to the definition given 
in Art. 16, the boiling point of a substance at a given pressure 
is the minimum temperature of the vapor of the substance at 
that pressure. This temperature is perfectly definite at a given 
pressure. The temperature at which water boils, however, in 
the ordinary sense of that term (meaning the formation of bubbles 
of steam near the bottom of the vessel) is slightly variable, it de- 
pends to some extent upon the rapidity with which heat is given 
to the boiling water and to some extent upon cleanness of the ves- 
sel. The connection of the phenomenon of boiling with what 
is stated above concerning the minimum temperature of a vapor 

*For more extensive tables see Castell-E vans' Physico- Chemical Tables, or 
Landolt & Bornstein's Physikalisch-Chemische Tabellen. 
fSee discussion of evaporation versus boiling in Art. 25. 



water at =. 

atmo8pheric ~— 

pressure ~ 




SOLIDS, LIQUIDS AND GASES. 321 

at a given pressure may be explained as follows: Figure 16 
represents a bubble of water vapor or steam 5 underneath water 
at atmospheric pressure. Therefore the steam itself must be at 
atmospheric pressure,* and if its tempera- 
ture (the temperature of the water) is less 
than that for which steam can exert one 
atmosphere of pressure, the bubble of 
steam will condense into liquid and col- 
lapse. The temperature of the water must 
be at least as great as the minimum tem- 
perature of water vapor at atmospheric ~K 
pressure in order that the bubble of steam 
may continue to exist, and if the tempera- 
ture of the water is slightly greater than Fig l6 
this the bubble of steam will continue to 
grow in size as more steam is formed at its boundaries. 

18. Variation of boiling and freezing points with pressure. 

As defined in Art. 16 the boiling point of a substance at a given 
pressure is the minimum temperature of the vapor of that sub- 
stance at the given pressure. This temperature varies greatly 
with the pressure as exemplified in the above tables for water, 
and anhydrous ammonia. This variation of boiling point with 
pressure is utilized in the ammonia refrigerating machine. The 
essential features of this machine are shown in Fig. 17. Liquid 
ammonia vaporizes in the cooling pipes at a low pressure and 
low temperature, taking in heat from the surroundings. The 
pressure is maintained at a low value by means of a pump which 
removes the ammonia vapor from the cooling pipes and com- 
presses it into condensing pipes as shown in the figure. The 
ammonia vapor is converted into liquid form in the condensing 
pipes, giving off heat to the surroundings, and the liquid ammonia 
is then allowed to flow back into the cooling pipes where it is 
again vaporized at low pressure and low temperature. If the 

*A bubble of steam near the bottom of a deep vessel of boiling water is, of course, 
above atmospheric pressure. 
22 



322 



THEORY OF HEAT. 



pressure in the cooling pipes is kept at, say, 1.14 atmospheres 
the temperature of the boiling liquid ammonia will be about 
30 below zero centigrade, according to the table on page 






condensing 
water 



NH vapor 



i=Th± 




NH vapor 



valve 



NH 3 liquid ^ g 



valve 



NH liquid 

>2 



condensing pipes 

temperature T. 

(hot) 



cock 



Fig. 17. 



cooling pipes 
temperature T 
(cold) ' 



320, and if the pressure in the condensing pipes is kept at, 
say, 25.6 atmospheres, the vapor will condense at about 6o° 
centigrade. 

Regelation. The melting point of ice is lowered by pressure as 
stated in Art. 16. This is strikingly shown by the following 
experiment: A block of ice is supported on two pillars and a 
fine steel wire with a heavy weight on each end is hung over it. 
Where the wire rests against the ice the pressure is considerable 
and the melting point of the ice is low, say, one degree below 
zero centigrade. Some of the ice therefore immediately melts and 
causes * the temperature under the wire to fall to, say, one degree 

*No solid can exist as a solid above its melting point. If one attempts to warm 
a solid above its melting point, a portion of the solid melts and the heat which is 
given to the substance goes to produce the change of state without causing a rise 
of temperature. When a block of ice at o°C. is subjected to pressure, the melting 
point is lowered and the ice is therefore temporarily above its melting point. The 
result is that a portion of the ice melts and the heat used to produce the change of 
state cools the ice to its melting point. 



SOLIDS, LIQUIDS AND GASES. 323 

below zero at which point the temperature under the wire stands 
as long as the pressure is maintained. The water from the melt- 
ing ice flows around to the top of the wire where it is relieved of 
pressure and where it cannot, therefore, be cooler than o°C. 
The result is that heat continually flows from the warm region 
(at o°C.) on top of the wire into the cool region (at — i°C.) 
under the wire, the ice under the wire continues to melt, and the 
water as it flows around to the top of the wire continues to freeze. 
In the course of an hour or more a wire may thus cut its way 
through a large block of ice, leaving the block as one solid piece 
by the freezing of the water above the wire. 

This melting of ice at a point where it is subjected to pressure 
and the immediate freezing of the resulting water when it flows 
out of the region of pressure, is called regelation. The remark- 
able ease with which a skater glides over the ice is due in large 
part to the formation of a thin layer of water in the region of 
excessive pressure under the skate runners. This water of course 
freezes almost instantly when the skate has passed and the 
pressure relieved. The cohesion of particles of ice when pressed 
tightly together, as exemplified in the packing of snow in balls, 
is due to the melting of the ice particles at the points of contact 
where the pressure is great, and the immediate freezing of the 
resulting water as it flows out of the small regions of pressure. 
Snow must be nearly at o°C. in order that the phenomenon of 
regelation may be brought about by the slight pressure that can 
be produced by the hand. 

19. Boiling points and melting points of mixtures. When a 
moderately dilute solution of common salt in water freezes, pure 
ice is formed, and the whole of the salt is left in the residual 
liquid. When a solution of salt in water boils, pure water vapor 
is formed, and the whole of the salt is left in the residual liquid. 
In every such case, namely, when the dissolved substance does 
not pass off with the water vapor or crystallize with the ice, the 
boiling point of the solution is higher than the boiling point of 



324 



THEORY OF HEAT. 



pure water and the freezing point of the solution is lower than 
the freezing point of pure water.* 

When a weak solution of salt in water is frozen, pure ice 
freezes out of the solution, the residual liquid becomes richer and 
richer in salt, and the freezing point decreases more and more 
until the residual liquid begins to freeze ; and during the freezing 
of the residual liquid the freezing point does not change in value. 
When a very strong solution of salt in water is cooled, the solution 
"freezes" by the deposition of crystals of salt, the residual liquid 
becomes less and less rich in salt, and the freezing point lowers 
more and more until the residual liquid begins to freeze as before. 



7* 

1 
I 



















^4^\ 




Bl 








J 








C 





10 20 30 

^percentage of salt 

Fig. 18, 



40 



The ordinates of the curve ABC, Fig. 18, represent the freezing 
points of solutions of salt (NaCl) in water. At any point on the 
branch A C of the curve, the freezing consists in the deposition 



*A good discussion of this subiect may be found in Whetham's Theory of Solution, 
Cambridge University Press, 1902. It is also discussed at considerable length in 
Nernst's Theoretical Chemistry (English translation Macmillan & Co., London, 
1904), and in Jones's Physical Chemistry (The Macmillan Co., New York, 1902). 



SOLIDS, LIQUIDS AND GASES. 



325 



of ice crystals, and at any point on the branch B C the "freezing " 
consists in the deposition of crystals of salt. The point C of 
minimum freezing temperature is called the eutectic point, and 
the residual liquid which freezes at this temperature is called 
the eutectic mixture of salt and water. A microscopic examination 
of the frozen eutectic mixture of water and salt reveals the exis- 
tence of minute crystals of pure ice and minute crystals of pure 
salt side by side. 

The phenomena of freezing of fused mixtures of salts and the 
phenomena of freezing of metallic alloys are similar to the 

per cent of lead 





go 80 70 60 5° 4o »3 


20 10 


35° 

300 

as 

5250 
2 


^i 


























































B, 


1 












« 










































1: 









10 20 30 40 50 60 70 80 90 100 
per cent of tin 

Fig. 19. 

phenomena of freezing of salt solutions, and the above described 
behavior of a solution of common salt is perhaps the simplest 
case. Thus Heycock and Neville* have found many cases in 
which the freezing of a metallic alloy causes the deposition of pure 
crystals of one metal or the other, in the same way that the 
freezing of a salt solution causes the deposition of pure crystals 
of ice or pure crystals of salt. The freezing point of the residual 
alloy is steadily lowered by the deposition of pure crystals of 

^Journal of Chemical Society since 1888. See Nernst's Theoretical Chemistry 
(Macmillan & Co.), page 402. 



3^6 THEORY OF HEAT. 

either metal until a certain point (the eutectic point) is reached 
when the residual alloy (the eutectic alloy) continues to freeze 
without further drop of temperature. 

Another case which is slightly more complicated is exempli- 
fied by alloys of lead and tin, of which the freezing-point dia- 
gram is shown in Fig. 19.* The alloy of the two metals which 
has the minimum freezing point is called the eutectic alloy. Along 
the branch A C of the curve, crystals of lead are deposited con- 
taining a variable percentage of tin ranging from pure lead at A 
up to 12 atoms of tin to 88 atoms of lead as the eutectic point C 
is approached; along the branch BC of the curve, crystals of 
tin are deposited containing a variable percentage of lead ranging 
from pure tin at B up to one atom of lead to 500 atoms of tin 
as the eutectic point is approached. The crystals of lead 
containing a variable percentage of tin and the crystals of 
tin containing a variable percentage of lead are called solid 
solutions. 

Figure 20 is a melting-point diagram of alloys of copper and 
magnesium, f These alloys present three eutectic points as indi- 
cated in the figure and four so-called distectic points, or points 
of maximum freezing temperature. The first third of this dia- 
gram, between distectic points 1 and 2, is a melting-point dia- 
gram of mixtures of pure magnesium and the chemical compound 
Mg 2 Cu; the middle portion of the diagram, between distectic 
points 2 and 3, is a melting-point diagram of mixtures of the 
two chemical compounds Mg 2 Cu and MgCu 2 ; and the last 
third of the diagram, between distectic points 3 and 4, is a melt- 
ing-point diagram of mixtures of the chemical compound MgCu 2 
and pure copper. 

When a cast metal is slowly cooled, the outside portions of the 
casting differ very considerably in composition from the interior 
portions of the casting; any substance which is present in the 

*Taken from a paper by W. Rosenheim and P. A. Tucker, Philosophical Trans- 
actions, of the Royal Society, Series A, Vol. 209, pages 89-122, November 17, 1908. 

fFrom a paper by G. Urazov, abstracted in the Chemische Centralblatt, page 
1038 for the year 1908. 



SOLIDS, LIQUIDS AND GASES. 



327 



atoms per cent of Mjj 

$6 80 !7o i6o 50 40 30 20 to 




30 40 50 60. 70 bo 90 loo- 
atoms per cent of -Cu 
Fig. 20. 

metal in small quantity tends to collect in the central parts of 
the casting.* 

The use of ice and salt as a freezing mixture. Ice in a strong 
solution of common salt has a very low melting point, 15 or 20 
degrees below zero centigrade (see Fig. 18). Therefore ice mixed 
with salt falls to a temperature of 15 or 20 degrees below zero 
centigrade, and stands at that temperature (if there is an excess 

*A good introduction to the study of metallic alloys is Ibbotson's translation 
of Goerens' Introduction to Metallography, Longmans, Green & Co., 1908. 



3^S THEORY OF HEAT. 

of undissolved salt) until all of the ice is melted by heat absorbed 
from surrounding objects. A vessel of pure water or cream 
surrounded by a mixture of ice and salt gives off heat to the 
very cold mixture until the water or cream is frozen. The 
sprinkling of salt on ice or snow in the winter time does not, as 
commonly supposed, melt the ice; it lowers the melting point 
below the temperature of the surroundings (if this is not more 
than 15 or 16 degrees below zero centigrade) and the ice is melted 
by the heat abstracted from its surroundings. 

It should be especially noted that the action of the salt on the 
water is not the cause of the intense cold. Thus the combination 
of sulphuric acid and water causes a generation of heat and con- 
sequently a rise of temperature, whereas a mixture of ice and 
moderately dilute sulphuric acid falls to a very low temperature. 
The freezing point of the ice is lowered in the presence of acid or 
salt, some of the ice melts in consequence of this fact, and the 
heat which produces the melting is absorbed from the surrounding 
ice and solution, thus producing a fall of temperature. 

Many substances cause a drop of temperature (an absorption 
of heat) when they are dissolved in water even when no ice is 
present. Thus every photographer is familiar with the very per- 
ceptible cooling which is produced when "hypo" (sodium hypo- 
sulphite) is dissolved in water. 

20. Latent heat of fusion and latent heat of vaporization.* 

When heat is imparted to a substance which is at its melting point 
or at its boiling point, a portion of the substance is melted or 
vaporized and the temperature remains unchanged. The number 
of thermal units required to change unit mass of the solid sub- 
stance at its melting point into liquid at the same temperature 
is called the latent heat of fusion of the substance. The number 
of thermal units required to change unit mass of a liquid f at its 

*Berthelot's method for determining the latent heat of steam is quite fully de- 
scribed on pages 152 and 153 of Edser's Heat for Advanced Students. Methods for 
determining latent heats of fusion and latent heats of vaporization are discussed 
in every physical laboratory manual. 

fin some cases the substance changes directly from the solid form to vapor 
without passing through the liquid state. This matter is discussed in Art. 23. 



SOLIDS, LIQUIDS AND GASES. 



329 



boiling point into vapor at the same temperature is called the 
latent heat of vaporization of the liquid. 

The boiling point of a substance varies greatly with pressure, 
and the latent heat of vaporization varies greatly with the tem- 
perature of the boiling point of the given substance. Thus, the 
latent heat of vaporization of water is 1043 British thermal units 
per pound at a pressure of one pound per square inch (absolute) 
and at a temperature of io2°F., it is 965.7 British thermal units 
per pound at standard atmospheric pressure and 2i2°F., and it 
it 844.4 British thermal units per pound at 200 pounds per 
square inch (absolute) and 38i°.6F. 

The following table gives freezing and boiling points and latent 
heats of fusion and vaporization of a number of substances. 

TABLE. 



Melting 
point. 



Water 

Alcohol 

Lead 

Mercury. . . . 
Sulphur 

Ether 

Carbon 

bisulphide 
Chloroform . 



o°C. 

327 
-39-5 

115 



Latent heat ot 

fusion, calories per 

gram. 



80 



5-86 
2.82 
9-36 



Boiling point at 

atmospheric 

pressure. 



I00°C. 

73,3 

1500. about 
357- 
444-7 
34-9 

46.8 
61. 1 



Latent heat of 

vaporization, calories 

per gram. 



536 
209 

62 

91 

86.6 
58.5 



Total heat of steam. The amount of heat required to raise water from a chosen 
standard temperature, say o°C, to the boiling temperature at a given pressure and 
to convert it into steam at that temperature and pressure is called the "total heat'" of 
the steam at that temperature and pressure. According to Regnault's experiments, 
the total heat of steam is given with sufficient accuracy for most practical purposes 
by the equation 

# = 606.5+0.305* 

in which H is the total number of calories required to raise the temperature of one 
gram of water from o°C. to t°C. and convert it into steam at that temperature, the 
pressure of course being such as to cause the water to boil at temperature f.* 

*Regnault's experiments are very briefly described in Edser's Heat for Advanced 
Students, pages 1 53-155- Very extensive tables of the total heat of steam are given 
in treatises on the steam engine. 



33° THEORY OF HEAT. 

21. Superheating* and undercooling of liquids. When water 
which is free from air and dust is heated in a clean glass vessel, 
its temperature is likely to rise 10 degrees or more above its 
boiling point (corresponding to the given pressure); and when 
it begins to boil it does so with almost explosive violence and the 
temperature quickly falls to the boiling point. If pure water is 
cooled in a clean glass vessel, its temperature is likely to fall 
considerably below its normal freezing point; and when freezing 
begins a large amount of ice is suddenly formed and the tem- 
perature quickly rises to the normal freezing point. It seems 
that water cannot change to vapor or to ice except there be some 
nucleus at which the change may begin. Most liquids show 
these phenomena of superheating and undercooling. 

22. Critical states. When a liquid and its vapor (confined in a vessel) are 
heated, a portion of the liquid vaporizes, the pressure is increased, the density of 
the vapor increases and the density of the liquid decreases.! When a certain tem- 
perature is reached, the density of the liquid and the density of the vapor become 
equal and the vapor and liquid are identical in their physical properties. This 
temperature is called the critical temperature of the liquid, the corresponding pres- 
sure is called the critical pressure, and the corresponding density is called the criti- 
cal density. The heat of vaporization of a liquid is less the higher the temperature 
(and pressure) at which vaporization takes place and it becomes zero at the crit- 
ical temperature. J 

23. Sublimation. At a very low pressure the vapor of a given 
substance must be cooled to a very low temperature to produce 

*The term the superheating of a liquid must not be confused with the term the 
superheating of steam. Superheated steam is unsaturated steam, that is, steam of 
which the pressure is less than the maximum pressure at the given temperature, 
or of which the temperature is greater than the minimum temperature for the given 
pressure. Steam may be superheated by passing saturated steam from a steam 
boiler through a coil of pipe in a furnace. 

fThis statement may not be exactly correct in some cases. The density of 
liquid and vapor become more and more nearly equal in every case. 

JA good discussion of the subject of critical temperatures and pressures including 
the celebrated experiments of Andrews on carbon dioxide is given in Edser's Heat 
for Advanced Students, pages 201-219. An introducing to van der Waal's theory 
of corresponding states is given in Edser's Heat for Advanced Students, pages 304- 
314. A very full discussion of van der Waals's theory of corresponding states is 
given in Nernst's Theoretical Chemistry, pages 224-230. Macmillan & Co., Lon- 
don, 1904. 



SOLIDS, LIQUIDS AND GASES. 33 1 

condensation. If this temperature is below the freezing point 
of the substance, then the vapor will be condensed in solid form 
without passing through the intermediate liquid stage. Thus, 
to condense the slight amount of water vapor which is in the air 
in the winter time, the temperature of the air sinks far below the 
freezing point and the moisture is condensed in the form of 
snow-flakes or frost crystals. On the other hand, if the amount 
of moisture in the air on a cold winter's day is extremely small, 
snow and ice evaporate slowly without passing through the inter- 
mediate liquid stage. The word sublimation was applied by the 
early chemists to the process of distillation in which a solid is 
converted directly into a vapor and the vapor condensed into 
a solid. The most familiar example of this process of sublimation 
is that which is furnished by gum camphor. Every one perhaps 
has observed the formation of fine crystals of gum camphor on the 
cold side of a stoppered bottle which contains lumps of gum cam- 
phor. The gum camphor is converted into vapor in the warmer 
parts of the bottle and the vapor is condensed into crystals in 
the cooler parts of the bottle. The same phenomenon takes 
place in closed bottles containing crystals of iodine. 

24. Pressures of mixed gases. When two or more gases are 
mixed in a vessel, the total pressure is equal to the sum of the 
pressures which each component gas would exert if it occupied 
the vessel alone {Dalton's Law). For example, if the amount 
of air in a vessel is such that it alone would exert a pressure p 
and if the amount of water vapor in the vessel is such that it 
alone would exert a pressure w, then the mixture will exert a 
pressure p + w.* 

A result of Dalton's Law is that a definite portion of the total 
pressure of a mixed gas may be considered to be due to each of 
the component gases of which the mixture is made. Thus the 
total pressure of the atmosphere is due in part to the nitrogen, 
in part to the oxygen, in part to the carbon dioxide, in part to 

*This statement is not exactly true, the degree of approximation being about 
the same as in the case of Boyle's Law and Gay Lussac's Law. 



33 2 THEORY OF HEAT. 

the water vapor, in part to the argon, etc., of which the atmosphere 
is a mixture. 

25. Evaporation versus boiling. It is a common observation 
that water evaporates into the air at temperatures far below 
ioo°C. A liquid at a given temperature continues to evaporate so long 
as the pressure of its vapor is less than the maximum pressure its 
vapor can exert at the given temperature. This is true whether 
the space above the liquid is filled with vapor alone or with 
vapor mixed with any gas at any pressure. For example, water 
vapor can exert a pressure of 355 millimeters. of mercury at 8o°C. 
and if a vessel at 8o°C. contains water, the water will vaporize 
until the pressure of the water vapor in the vessel is 3 5 5 millimeters. 
If the vessel contains nothing but water vapor then, of course, the 
total pressure will be 355 millimeters when equilibrium is reached. 
If the vessel contains dry air at atmospheric pressure, some of 
the air will be driven out by the vapor which is formed, and when 
equilibrium is reached the water-vapor pressure in the vessel will 
be 355 millimeters and the air pressure will be 405 millimeters, 
making a total of 760 millimeters.* If the vessel is filled with 
dry air at any pressure p and suddenly closed before any per- 
ceptible amount of water vapor is formed, then water vapor will 
form until the total pressure is p -f- 355 millimeters, p being the 
pressure due to the air alone and 355 millimeters being the pres- 
sure of the water vapor. 

26. Atmospheric moisture. Hygrometry. Dew Point. The 
dew point is the temperature to which the atmosphere must be 
cooled in order that the water vapor which is present may be 
saturatedo Further cooling of the atmosphere would cause some 
of the moisture to condense. 

Vapor pressure. That part of the pressure of the atmosphere 
which is due to the water vapor which is present is called the 
vapor pressure. This pressure varies from nearly zero to 30 
millimeters, or more. 

*The outside air pressure is assumed to be 760 millimeters. 



SOLIDS, LIQUIDS AND GASES. 333 

Absolute humidity. The amount of water in the air, usually 
expressed in grams of water per cubic meter of air, is called the 
absolute humidity of the air. The absolute humidity varies 
from one gram of water, or less, per cubic meter of air on a very 
cold, dry winter's day to 30 or 35 grams of water per cubic meter 
of air on a moist summer's day. 

Relative humidity. The amount of water in the air expressed 
in hundredths of what the air would contain if it were saturated 
at the given temperature is called the relative humidity. When 
the relative humidity is low, the air is said to be dry; when the 
relative humidity is high, the air is said to be moist, irrespective 
of the actual amount of water which is present. For example, 
20 grams of water per cubic meter would correspond to a relative 
humidity of about 60 per cent, on a warm summer's day and the 
air would seem to be extremely dry, whereas about 5 grams of 
water per cubic meter of air would saturate the air at o°C. and 
the air would seem extremely moist. 

The method usually employed for the determination of the 
hygrometric elements (dew point, pressure of vapor, absolute 
humidity, and relative humidity) is by use of wet and dry bulb 
thermometers, from the readings of which the various quantities 
may be determined from empirical tables. Such tables are 
published by the United States Weather Bureau.* 

27. Dissociation pressures. A phenomenon which is analo- 
gous to evaporation is the dissociation of a solid or a liquid sub- 
stance by heat, a portion of the substance being given off in the 
form of vapor and the remainder being left in the form of a 
solid or liquid. An example will serve best to make the matter 
clear. Calcium carbonate (ordinary limestone) dissociates into 
calcium oxide (ordinary lime) and carbon dioxide when it is 
heated. If, however, calcium carbonate is brought to a given 
temperature in a closed vessel, the dissociation is arrested when 
the carbon dioxide reaches a definite pressure which is called the 
dissociation pressure of the calcium carbonate at the given tem- 

*Weather Bureau Bulletin No. 235. Price, 10 cents. 



334 



THEORY OF HEAT. 



perature. An increase of pressure or decrease of temperature 
casues some of the carbon dioxide to recombine with the free 
calcium oxide, and a decrease of pressure or increase of tempera- 
ture causes some of the unchanged calcium carbonate to dis- 
sociate. 

28. Transition temperatures. A crucible containing melted 
zinc is removed from a furnace and allowed to cool. The abscis- 

f6c 



1500 



1400 



[30c 



1 iod 



.g ,io'oo 

'1 

I 900 



800 



700 



600 



500 



\fluid 


iron 

freezing point 1505°C 
























































\t> 


ron hoi 


id) 


















































































\/3-iri 


nholit 


') 








880°C 












780' C 


























&- iroi 


i(soUd] 


1 























20 30 40 50 60 70 80 90 100 

minutes 

Fig. 21. 



sas of the curve in Fig. 13 represent elapsed times and the ordi- 
nates represent observed temperatures of the zinc. The tem- 
perature of the metal drops steadily until it begins to freeze at 



SOLIDS, LIQUIDS AND GASES. 



335 



a temperature of 4i9°C, the temperature then remains constant 
until all of the zinc is frozen, after which the temperature again 
drops steadily. The curve in Fig. 13 is called a cooling curve. 
Fig. 21 represents a cooling curve of a crucible containing 
pure melted iron. In this case the freezing of the iron is indicated 
1300 



TOOO 



v 900 
I 

H 

t 800 



700 



600 



500 






30 



40 



50 



minutes 

Fig. 22. 

by a stationary temperature, and two other changes of state of 
the iron are indicated by stationary temperatures at 88o°C. 
and 78o°C. respectively. Solid iron seems to exist in three modi- 
fications which are called a-iron, /3-iron and 7-iron, respectively. 
These modifications of solid iron differ in their crystalline struc- 
ture. A remarkable physical property of a-iron is that it is 



336 



THEORY OF HEAT. 



highly magnetic whereas jS-iron and 7-iron are non-magnetic. 
The two temperatures 78o°C. and 88o°C. are called transition 
temperatures of iron. 

The admixture of other substances affects not only the freezing 
point of a substance but also its transition temperatures. Thus 
cast iron is an alloy of iron with about five or six per cent, of 
carbon, and the cooling curve of cast iron is shown in Fig. 22. 
The freezing in this case begins at ii5o°C. and the freezing point 
is steadily lowered to about H20°C. as the freezing progresses. 



150 



S3 
Hi so 



591 

5, JO 15 20 25, 30 35 40 45 5° 

minutes 

Fig. 23. 

Below iioo°C. the iron is solid and the effect of the carbon is 
to lower both of the transition temperatures, which are shown in 
Fig. 21, to about 7oo°C. Thus cast iron has apparently but one 
transition temperature. 

The recalescence of steel. The full-line curve in Fig. 23 is 
the cooling curve of water, the stationary temperature ab being 
that which occurs during condensation, and the stationary tem- 
perature cd being that which occurs during freezing. If the 
water is very pure it cools considerably below the freezing point 























\j 


earn 


















\l 




condens 


ation 












t 


b 
























\w 


ater 


2 


d 










freezing 




/S 


















N.-.' 




x% 


















ice N 


"^ 



SOLIDS, LIQUIDS AND GASES. 337 

before freezing begins, as shown by the dotted line. When the 
temperature falls below the freezing point in this way a sudden 
rise of temperature takes place when freezing does begin, as 
shown by the dotted line. A phenomenon somewhat similar 
to this phenomenon of under-cooling of water occurs in cast 
iron and steel. When the cast iron or steel is cooled, the change 
from one modification to the other takes place after the temper- 
ature has fallen below the true transition temperature, and when 
the transformation does begin the temperature suddenly rises. 
This sudden rise of temperature is called recalescence, and it may 
be seen by heating a piano steel wire by an electric current and 
watching it as it cools; when it reaches a low red heat a sudden 
flash of brighter redness occurs after which the temperature 
again falls. 

Retarded transformations. The hardening of steel. The phe- 
nomenon of the under-cooling of a liquid which is described in 
Art. 21 may be thought of as a slightly retarded transformation. 
The transformation from pure water to ice does not begin exactly 
at o°C. when the water is cooled but at a lower temperature, and 
when the transformation does begin the mixture rises quickly to 
the normal freezing point. A familiar example of an almost 
permanently retarded transformation is that which is afforded 
by molasses candy. The crystallization of syrup is a very slow 
process because, apparently, of the viscosity of the syrup. If the 
syrup is cooled very, very slowly the crystallization takes place 
at the true freezing temperature, but if the syrup is cooled quickly 
it does not have time to crystallize, and the result is the well 
known molasses candy. Paradoxical as it may seem, the syrup 
when it is cooled suddenly does not have time to "freeze" but 
remains in that physical modification which is the stable modi- 
fication at high temperatures. If, however, molasses candy is 
allowed to stand for some months the "freezing" gradually 
comes about, transforming the substance into the crystalline 
modification which is stable at low temperatures. 
23 



33 8 THEORY OF HEAT. 

Retarded transformations take place in the hardening of steel 
and are of very great importance as follows. At a high tempera- 
ture steel settles to thermal equilibrium with a certain crystalline 
structure, that is, with the iron in a certain modification and with 
certain crystalline compounds of iron and carbon present. If 
the steel is very slowly cooled the various transformations take 
place at approximately the true transition temperatures, and 
we have what is called annealed steel which is the stable form of 
steel at low temperatures. If, however, the steel is cooled very 
quickly the transformation from one modification to another 
does not have time to take place, the form or modification of the 
steel which normally exists and is stable at high temperatures is 
left in existence at ordinary temperatures, and we have the fam- 
iliar hard form of steel. Hardened steel is an unstable modifica- 
tion and it tends gradually to change to the stable modification 
(soft annealed steel).* This change is greatly hastened by a 
slight rise of temperature. Thus hard steel is tempered by heat- 
ing it slightly for a short time. 

29. Coexistent phases. The forms of a substance which can exist together in 
thermal equilibrium are called coexistent phases of that substance. Thus the water 
in a vessel at a given temperature may be partly liquid and partly vapor (a liquid 
phase and a vapor phase) ; or the water in a vessel may be partly liquid and partly 
ice (a liquid phase and a solid phase). Water vapor may be wholly converted into 
water, and water may be wholly converted into ice, and vice versa; therefore these 
phases are said to be phases of the same composition. 

A mixture of ice and salt in a vessel may be partly in the form of solution and 
partly in the from of crystals of salt, thus presenting a liquid phase and a solid 
phase. The liquid and the crystals in this case are phases of different composition 
inasmuch as a salt solution cannot be converted wholly into salt, nor can salt alone 
be converted into salt solution. 

This breaking up of substances into phases of different composition is the fun- 
damental fact of chemistry. Thus, when solutions of silver nitrate and sodium 
chloride are mixed, the mixture settles to thermal equilibrium with a solid phase 
consisting of precipitated silver chloride and a liquid phase consisting of a solution 
of sodium nitrate. When a mixture of alcohol and water is evaporated, the liquid 
and vapor both contain water and alcohol, but the percentage of alcohol is much 
greater in the vapor than in the liquid. Thus, if a weak solution of alcohol be partly 

*Extremely hard steel gradually softens during the course of a number of years 
See a paper by Carl Barus, Physical Review, Vol. XXIX, pages 516-524, December, 
1909. 



SOLIDS, LIQUIDS AND GASES. 339 

distilled, the greater part of the alcohol passes over in the first distillate and the 
greater part of the water remains behind. This process is called fractional dis- 
tillation. 

30. Elementary and compound substances. A substance which can be broken 
up into phases of different composition is called a compound substance. Thus a salt 
solution is a compound substance and a mixture of alcohol and water is a compound 
substance because both can be separated into phases of different composition, water 
and salt in the one case and water and alcohol in the other case. The component 
parts of a given compound substance may themselves be compound. Thus, it is 
possible to separate water into phases of different composition. The simplest 
method for doing this is to pass an electric current through the water when two 
gases (oxygen and hydrogen) having entirely different properties are obtained. 
Substances which have never yet been broken up into phases of different composi- 
tion are called elementary substances or chemical elements. For example, oxygen, 
hydrogen, iron, lead, sulphur, etc., are chemical elements. 

31. Chemical compounds; mixtures. The component parts of some compound 
substances are always in unalterably fixed proportions. Thus, two volumes of 
hydrogen always combine with one volume of oxygen to produce water; 23.05 
parts by weight of sodium always combine with 35.45 parts by weight of chlorine 
to form sodium chloride, and so on. Such compound substances are called chemical 
compounds. On the other hand, there are many compound substances of which 
the component parts may be in widely varying proportions; such compound sub- 
stances are called mixtures. Thus, alcohol and water may be mixed in any pro- 
portion. Two substances are said to combine when they form a chemical compound 
and they are said to mix when they form a mixture. There is no very sharp line of 
division between these two processes, in general when substances combine (in fixed 
proportions to form a chemical compound) a large amount of heat is usually de- 
veloped whereas when two substances mix (in variable proportions to form a mix- 
ture) a very small amount of heat is usually developed. The sharpest line of div- 
ision between chemical compounds and mixtures, however, is that a chemical com- 
pound freezes or evaporates at a fixed temperature, pressure being constant. Thus, 
if water is cooled until it begins to freeze, the temperature will remain constant until 
it is all frozen, and if water is heated until it begins to boil, the temperature will 
remain constant until it is all vaporized. The freezing point of a mixture, however, 
slowly decreases as more and more of the mixture is frozen and the boiling point of 
the mixture slowly rises as more and more of the liquid is vaporized.* 

32. Combining ratios. Law of constant proportions. Law of multiple pro- 
portions. Extremely fine iron filings may be burned in the air without any of the 
products of the combustion being scattered as is the case in the burning of coal. 
This burning of the iron filings is the combination of the iron with the oxygen of 
the air, the increase of weight after burning is the weight of the oxygen which has 
combined with the iron, and the weight of the oxygen and the weight of the iron are in 
a fixed ratio to each other. This is true of all chemical combinations, as stated in 
the previous article, and it is called the law of constant proportions. 

*This statement is true in the great majority of cases, but it is not universally 
true. 



340 THEORY OF HEAT. 

The ratio of the weight of iron to the weight of oxygen which combines with it is 
called the combining ratio of these two elements. Two elements often have more 
than one combining ratio. Let b x , b 2 , or & ;J , be the masses of one element which can 
combine with a given mass a of another element. The masses b lt b 2 , b z are always 
multiples of some one number, that is to say, the ratios b x : b 2 , b 2 : & 3 , etc., are rational 
fractions. This fact is called the law of multiple proportions. An illustration of 
this law is given in Art. 34 where the compounds of nitrogen and oxygen are 
described. 

33. Chemical combination of gases. Let u and v be the respective volumes of 
two gases (measured at the same temperature and pressure) which unite to form 
a chemical compound. The ratio u/v is always a rational fraction.* 

Thus two volumes of hydrogen combine with one volume of oxygen to form 
water. Two volumes of carbon monoxide (CO) combine with one volume of oxygen 
to form C0 2 , equal volumes of hydrogen and chlorine combine to form HC1. Ni- 
trogen and oxygen may combine in the following proportions by volume: 2 of 
nitrogen with 1 of oxygen; 1 of nitrogen with 1 of oxygen; 2 of nitrogen with 3 of 
oxygen; 1 of nitrogen with 2 of oxygen; and 2 of nitrogen with 5 of oxygen. 

34. The molecular theory. The above facts of chemical combination are 
clearly represented to our minds if we assume that each chemical element is made 
up of similar particles of equal mass called atoms, and that the atoms of two or more 
elements in a chemical compound are arranged in similar groups called molecules. 
For example, the atomic groups or molecules of the five compounds of nitrogen and 
oxygen are as follows: 

Compound No. 1, 2 atoms of nitrogen and 1 atom of oxygen, N s O 

Compound No. 2, 1 atom of nitrogen and 1 atom of oxygen, NO. 

Compound No. 3, 2 atoms of nitrogen and 3 atoms of oxygen, N 2 3 . 

Compound No. 4, 1 atom of nitrogen and 2 atoms of oxygen, N0 2 . 

Compound No. 5, 2 atoms of nitrogen and 5 atoms of oxygen, N 2 5 . 
The combining ratios of these various compounds are 2802 : 1588, 1401 : 1588, 
2802 : 4764, 1401 : 3176, and 2802 : 7940, respectively. These numbers are mul- 
tiples of 1401 on the one hand and of 1588 on the other hand. The number 1401 
is called the atomic weight of nitrogen, and the number 1588 is called the atomic 
weight of oxygen, f 

35. The principle of Avogadro. According to the molecular theory a definite 
number of atoms of one gas unite with a definite number of atoms of another gas 
to form a molecule when the gases combine chemically. When gases combine 
chemically, however, a definite number of volumes of the one gas always combine 
with a definite number of volumes of the other as explained in Art. 3^. Therefore 
at an early stage of the development of the molecular theory, the hypothesis was 
advanced by Avogadro that all gases have the same number of molecules per unit 
volume at the same temperature and pressure. This hypothesis has been substantiated 

*This statement is not exactly true. The degree of exactness is of the same order 
as the degree of exactness of Boyle's Law and Gay Lussac's Law. 

•jThese are the atomic weights of nitrogen and oxygen when the atomic weight 
of hydrogen is arbitrarily chosen equal to 100. 



SOLIDS, LIQUIDS AND GASES. 34 1 

by every bit of experimental evidence which has been brought to bear upon it; it 
is now considered to be established and it is called Avogadro's principle. 

Problems. 

39. A copper vessel weighing one kilogram contains 12 kilo- 
grams of water at 30°C. Into this vessel are dropped at the 
same instant one kilogram of copper at ioo°C, 1.2 kilograms of 
zinc at 6o°C. and 1.5 kilograms of ice at — 20°C. Find the 
resultant temperature. The specific heat of ice is 0.51 and the 
latent heat of fusion of ice is 80 calories per gram. Ans. I7°.65 C. 

Note. The specific heat of copper and zinc are given in problems 32 and 33. The 
best method to adopt in the solution of such a problem as this is (1) to calculate 
the total amount of heat which would have to be taken from the mixture to bring 
everything to o°C. If this amount of heat is less than enough to melt the given 
amount of ice, the fractional part of the ice which can be melted thereby can be 
calculated and in this case the resultant temperature is o°C. with this fraction of 
the ice melted. If, however, the amount of heat which would have to be abstracted 
from the mixture to bring everything to zero is more than enough to melt all the 
ice then the amount required to melt the ice may be subtracted from the total 
amount and the rise in temperature produced in all the materials by the remainder 
of the heat may then be calculated. 

40. An open vessel contains 500 grams of ice at a temperature 
of — 20°C. and heat is imparted to the vessel at the rate of 10 
calories per minute. Plot a curve showing elapsed times as ab- 
scissas and temperatures of vessel as ordinates, assuming that 
the vessel gives no heat to surrounding bodies. The specific 
heat of ice is 0.51 and latent heat of fusion of ice is 80 calories per 
gram; the specific heat of steam is 0.38 and the latent heat of 
vaporization of steam at ioo°C. and normal atmosphere pres- 
sure is 537 calories per gram. 

41. Find the amount of heat required to raise 3 kilograms of 
lead at io° to its melting point and melt it. The mean specific 
heat of lead between io°C. and its melting point (325°C.) is 
about 0.035 and the latent heat of fusion of lead is 5.9 calories 
per gram. Ans. 50,775 calories. 

42. How much water at 5o°C. is required to melt 5 kilograms 
of ice at — io°C? Ans. 8.51 kilograms. 

43. A gas is collected over water at a temperature of i5°Cc 



342 THEORY OF HEAT. 

and the observed pressure is 752 millimeters. What would the 
pressure of the given amount of gas be if it occupied the same 
volume dry, that is, free from admixture of water vapor? Ans. 
739.33 millimeters. 

44. A gas is collected over water at a temperature of i8°C. 
The atmospheric pressure as determined by a barometer is 721 
millimeters. The pressure of the gas in the vessel is less than 
atmospheric pressure by an amount which is equivalent to a 
column of water 104 millimeters high. The observed volume 
of the gas is 645 cubic centimeters. What volume would the 
gas have if measured dry at o°C. and at a pressure of 760 milli- 
meters? Ans. 555 cubic centimeters. 

45. A closed bottle is full of dry air at 720 millimeters pressure 
and at a temperature of 5o°C. A small quantity of water is 
introduced into the bottle and the whole is allowed to stand 
until the water vapor is saturated throughout the enclosed space. 
What is the total pressure of air and water vapor? Ans. 811.98 
millimeters. 

46. How much ice per day (24 hours) would be required to 
reduce from 25°C. to ij°C. an air blast which furnishes one 
cubic meter per second, the air being measured at 760 millimeters 
and at a temperature of 25°C? Ans. 2028 kilograms. 

Note. The density of air at 760 millimeters and o°C. in 0.001293 grams per 
cubic centimeter, and the specific heat of air (at constant pressure) is 0.2375 calorie 
per gram. 

47. The heat of combustion of good anthracite coal is 7800 
calories per gram. A boiler is found by trial to evaporate 10 
kilograms of water at I20°C. per kilogram of coal burned, the 
temperature of the feed water being at 20°C. Find the fractional 
part of the heat of combustion of the coal which is utilized in 
the boiler. Ans. 81.5 per cent. 

Note. In this problem use the formula for total heat of steam on page 329. 



CHAPTER IV. 

THE ATOMIC THEORY OF GASES.* 

(Units of the c.g.s. system are used throughout this chapter except where it is 
explicitly stated to the contrary, that is to say, pressure is expressed in dynes per 
square centimeter, volume in cubic centimeters, mass in grams, etc., and tempera- 
ture is expressed in degrees centigrade. Absolute temperature is represented by T 
and temperature reckoned from the ice point is represented by I.) 

36. The gas laws. The principle of Avogadro which was reached at the end of 
the preceding chapter represents the conclusion to which chemists have been led 
concerning the nature of a gas. This conclusion is that a gas consists of a great 
number of small particles and that at a given temperature and pressure all gases 
contain the same number of particles per unit volume. This hypothesis as to the 
atomic f character of a gas gives a clear insight into many of the physical properties 
of gases, and the object of the present chapter is to develop this aspect of the 
atomic theory. For purposes of ready reference the various experimental facts 
concerning gases are here collected. 

Boyle s Law. The volume of a gas at constant temperature is inversely pro- 
portional to its pressure. 

Gay Lussac's Law. t (a) The pressure of a constant volume of a gas is propor- 
tional to the absolute temperature. 

(b) The volume of a given amount of gas at constant pressure is proportional to 
the absolute temperature. 

General formula for Boyle's and Gay Lussac's Laws. The complete relation 
between pressure, volume and temperature of a gas is expressed by the formula 

pv = MRT (4) bis 

in which p is the pressure of the gas, v is its volume, M is its mass in grams, T is the 
absolute temperature, and R is a proportionality factor . 

*What is here referred to as the atomic theory of gases is usually called the 
kinetic theory of gases. 

fThe terms atom and molecule refer to distinct ideas in chemistry. In physics, 
however, this difference is of little consequence except perhaps in some of the recent 
developments of spectrum analysis and in some of the recent studies of the discharge 
of electricity through gases. Therefore in this chapter the particles of a gas are 
called atoms or molecules indifferently. 

JThe statements here given of Gay Lussac's Law involve the experimental fact 
which is stated in Art. 5 together with definition of temperature ratios as given 
in Art. 6. The statements here given are slightly misleading inasmuch as they 
make it appear that absolute temperature has been previously and independently 
defined. 

343 



344 



THEORY OF HEAT. 



Law of integral volumes. Let u and v be the respective volumes of two gases 
(reckoned at the same temperature and pressure) which combine chemically; then 
the ratio u/v is always a simple rational fraction. 

Dalton's Law. A mixture of gases having no chemical action on each other 
exerts a pressure which is the sum of the pressures which would be exerted by each 
component gas separately if it occupied the containing vessel alone at the given 
temperature. 

Joule and Thomson's principle. When a gas escapes through an orifice 0, Fig. 
24, from a region CC of high pressure into a region DD of low pressure, both pres- 
sures being kept at constant values by proper movements of the pistons A and B, 



A i %igh pressure 



IT' 



mm 



W:i\''/r'\-;. 



I low pressure 



Fig. 24. 




porous plug 

Fig. 25. 



we have what is called free expansion. When a gas in a cylinder expands against 
a receding piston* we have what may be called constrained expansion. During 
constrained expansion (against a receding piston) a gas does work, as in a steam 
engine, the heat-energy contained in the gas decreases, and the temperature of the 
gas falls. The heat energy contained in a gas is not changed by free expansion, 
and the change of temperature as indicated by the thermometers T and T' in Fig. 
24 during the free expansion of hydrogen, nitrogen or oxygen was found by Joule 
and Thomson (Lord Kelvin) to be very small at ordinary temperatures and pressures. 

*It is important to notice that the moving piston A in Fig. 24 does not compress 
the gas in CC, it keeps its pressure and density constant; nor does the moving 
piston B allow the gas in DD to expand, it keeps its pressure and density constant. 



THE ATOMIC THEORY OF GASES. 345 

Joule and Thomson's observations were carried on by means of the arrange- 
ment shown in Fig. 25. The innumerable small passages in a porous plug served 
instead of an orifice, and the difference of pressure on the two sides of the porous 
plug was maintained by means of a pump, the moving pistons of which take the 
place of A and B in Fig. 24.* 

Joule and Thomson found that at ordinary temperatures and pressures hydrogen 
is slightly warmed by free expansion, whereas nitrogen and oxygen are slightly 
cooled. Gases having more complex molecules, for example, carbon dioxide (C0 2 ) 
and alcohol vapor (C 2 H e O) are cooled very considerably by free expansion. At 
very low temperatures and at very high pressures, all gases, as far as known, are 
very considerably cooled by free expansion. See Art. 38 on the liquid air machine. 

37. The kinetic theory of gases. Imagine a large number of very small moving 
particles (molecules) enclosed in a vessel. Imagine these particles to have the 
property of rebounding with undiminished velocity when they strike the walls of 
the vessel, to be so small as seldom to collide against each other, and to exert no 
perceptible attraction or repulsion on each other. Such a system of particles would 
exhibit all the properties of a gas. Therefore a gas is thought to consist of such a 
system of particles or molecules. 

Let p be the pressure of a gas, v the volume of the containing vessel (which is 
of course the volume of the gas), N the total number of molecules in the vessel, 
w(= N/v) the number of particles per unit volume, and m the mass of each mole- 
cule. Now, the kinetic energy of the system of particles is constant since the par- 
ticles rebound from the walls of the vessel with unchanged velocity. Therefore 
the average kinetic energy per molecule, namely, \mu 2 , is constant and definite in 
value; the quantity « 2 is the average value of the square of the velocities of the 
various particles; and we have: 

p = \nmu 1 (11) 

*A good description of Joule and Thomson's experiments is given in Edser's 
Heat for Advanced Students, pages 379-392. A very complete discussion of the 
theory of Joule and Thomson's experiments, including a proper consideration of 
the work done on the gas by piston A in Fig. 24, and the work done by the gas on 
piston B, is to be found in Buckingham's Thermodynamics, pages 127-137 (The 
Macmillan Company, 1900). Buckingham's discussion also includes the appli- 
cation of the results of Joule and Thomson's experiments to the question of the 
interpretation of the indications of an air thermometer. 

A discussion has arisen among physicists as to the cause of the decrease of tem- 
perature at the expansion nozzle in the liquid air machine (see Art. 38). This dis- 
cussion has given rise to an experimental study of the subject by Bradley and Hale 
whose results are published in the Physical Review, Vol. 29, pages 258-292. The 
experimental work of Bradley and Hale is interesting and important but it was 
not needed to settle the question under dispute. The cooling effect is due to free 
expansion. Indeed, a cooling effect due to free expansion in Fig. 24 leads to a state 
of affairs in which more work is done on the gas by piston A in Fig. 24 than is done 
by the gas on piston B and this excess of work tends to cause a rise in temperature 
of the gas so that the actual observed difference of temperature on the thermometers 
T and T' in Fig. 24 is less than that which corresponds to free expansion alone. 



346 THEORY OF HEAT. 

or, since n= N/v, 

pv = lNmo} 2 ( I2 ) 

Proof. The square of the velocity of a given particle is equal to the sum of the 
squares of the x, y and z components of its velocity. Therefore the sum of the 
squares of the velocities of all the particles is equal to the sum of the squares of 
all the ^-components, plus the sum of the squares of all the y-components, plus 
the sum of the squares of all the z-components. The particles move at random in 
all directions, so that the sum of the squares of the ^-components, of the y-com- 
ponents, and of the z-components are equal each to each. Therefore (i) The sum 
of the squares, Nofi, of all the velocities is equal to three times the sum of the squares 
of the x-components. 

Imagine the containing vessel to consist of two parallel walls, of area q, distant 
d from each other, perpendicular to the #-axis of reference, and between which the 
gas is confined. Only the x-components of the molecular velocities contribute, by 
impact, to the pressure on these walls, so that the y and z components may be ig- 
nored. Consider a single particle, the ^-component of whose velocity is a. This 
particle strikes first one wall and then the other, traveling back and forth a 1 2d 
times per second. At each impact the velocity of the particle changes by 2a, that 
is, from -j-a to —a, or the momentum of the particle changes by 2am. There- 
fore momentum is lost on each wall by the impact of this particle at the average 
rate 2amX,aJ2d, or ma 2 /d, which is the average force exerted on the wall by this 
particle. That is, the force on one wall, due to one particle, is equal to m/d times 
the square of its ^-velocity component. Therefore the total force F, exerted on 
the wall by all the particles, is equal to m/d times the sum of the squares of their 
^-velocity components. Therefore F = }/^oi 2 N • m/d; see (i). Dividing by q, and 
putting qd — v, we have F /q = p = }^Nm^ 2 /v. 

The kinetic theory of gases is very important as furnishing a clear conception 
of what constitutes thermal equilibrium of a gas, as furnishing a rational basis for 
Boyle's Law, Gay Lussac's Law, etc., and as enabling one to form clear mental 
pictures of various gas phenomena. 

Thermal equilibrium of a gas. When a gas is in thermal equilibrium, the erratic 
movements of its molecules are such that on the average there is the same number 
of molecules in each unit of volume of the gas and the same average molecular 
velocity in the neighborhood of each point in the gas.* 

Heating of a gas. When the walls of a containing vessel are heated the mole- 
cules of the enclosed gas rebound with increased velocity when they strike the walls 
and the temperature of the gas rises. When the walls of the containing vessel are 
cooled, the molecules of the gas rebound with diminished velocity when they strike 
the walls and the temperature of the gas falls. 

*An interesting simple discussion of the kinetic theory of gases, including van 
der Waal's theory, is given in Edser's Heat for Advanced Students, pages 287-314. 
The kinetic theory of gases forms one of the most important parts of mathematical 
physics. A good elementary treatise on the subject is Boynton's Kinetic Theory, 
The Macmillan Co., New York, 1904. See also Boltzman's Vorlesungen uber 
Gas Theorie. See also Planck, Acht Vorlesungen iiber theoretische Physik (Columbia 
University Lectures), Leipzig, 1910. 



THE ATOMIC THEORY OF GASES. 347 

Heating of a gas by compression. Cooling of a gas by expansion. When a gas 
is compressed under a piston in a cylinder, the particles of the gas rebound from 
the inwardly moving piston with unchanged velocity relative to the piston, but with 
increased actual velocity, and the temperature of the gas rises. When a gas is 
expanded under a receding piston in a cylinder, the particles of the gas rebound 
from the receding piston with diminished actual velocity and the temperature of 
the gas falls. 

Boyle's Law and Gay Lussac's Law. If we assume the absolute temperature of 
a gas to be proportional to the average kinetic energy per molecule, that is, if we 
assume T to be proportional to 3 / 2^ Ct,2 » we may write constant XMXT for }^Nmu> 1 
in equation (12), and this equation then becomes 

pv = MRT (4)bis 

in which R is a constant. On the basis, therefore, of the above assumption as to 
the relation between absolute temperature and average kinetic energy per mole- 
cule, the kinetic theory of gases is found to conform with Boyle's Law and Gay 
Lussac's Law. 

Avogadro's principle is shown to be consistent with the kinetic theory of gases 
as follows: Consider two gases and let p v n x , m v and « x be the pressure, number of 
particles per cubic centimeter, etc., of the one gas, and let p 2 , n 2 , m 2 , and w 2 be 
the corresponding quantities for the other gas. Then px = ]^n l m l w l i , and p 2 = 
}^n 2 m 2 w 2 2 > from equation (11). If the two gases are at the same pressure and 
temperature, then p x = p 2 and ww x 2 must be equal to m 2 w 2 2 according to the above 
assumption that absolute temperature is proportional to the average kinetic energy 
per molecule. Therefore when temperature and pressure are the same for the two 
gases, we have n x = n 2 . 

Dalton's Law is consistent with the kinetic theory of gases inasmuch as the 
moving particles are assumed to be so small that they do not interfere with each 
other in any way. Thus, if a number of oxygen molecules and a number of nitrogen 
molecules are in a containing vessel, each set of molecules will move exactly as if 
the others were not present and exert the same pressure as they would exert if they 
occupied the vessel alone. 

Joule and Thomson's principle. Consider a molecule of gas as it darts through 
the orifice in Fig. 24. This particle is moving away from the closely packed gas 
molecules in the high-pressure region and towards the more widely separated gas 
molecules in the low pressure region. Therefore, if there is an attractive force 
between the molecules of the gas, the molecule under consideration will lose velocity 
as it passes through the orifice because the backward force due to the attraction 
of the greater number of molecules behind it will exceed the forward force due to 
the attraction of the lesser number of molecules ahead of it. If, however, the mole- 
cules of the gas repel each other, a molecule will gain velocity when it darts through 
the orifice because the forward push of the greater number of molecules behind the 
given molecule will exceed the backward push of the lesser number of molecules 
ahead of it. 

The very small change of temperature of hydrogen, nitrogen and oxygen during 
free expansion shows that the molecules of these gases do not attract each other 
or repel each other to any great extent; in fact hydrogen molecules at ordinary 



34 8 THEORY OF HEAT. 

pressures and temperatures (when the molecules are relatively far apart) repel each 
other slightly ; nitrogen and oxygen molecules at ordinary pressures and tempera- 
tures attract each other slightly; and the molecules of all gases attract at low tem- 
peratures and pressures (when the molecules are relatively near together). 

38. Equation of van der Waals.* All gases deviate more or less from the laws 
of Boyle and Gay Lussac and show perceptible change of temperature when allowed 
to expand freely through an orifice, and a mixture of two gases does not exert a 
pressure which is exactly equal to the sum of pressures which would be exerted by 
the respective components of the mixture if they occupied the entire containing 
vessel alone at the given temperature. The molecules are not so small as seldom 
to collide, and the molecules do in general attract or repel each other. When the 
effects of collision and of attraction (or repulsion) are taken into account in the 
kinetic theory, an equation between pressure, volume and temperature is deduced 
which is more complicated than equation (12). This equation is arrived at as 
follows, and it is due to van der Waals. 

(a) Effect of size of molecules. If the moving particles have any size, collisions 
and impacts take place before the centers of the particles are coincident, or before 
the centers of the particles are in the plane of the wall of the containing vessel. 
Shorter distances are thus traversed between collisions, impacts are more frequent, 
and the pressure is greater than it would be if the particles were indefinitely small. 
The result is very much as if the volume of the containing vessel were smaller by a 
constant amount, b, than it really is. Equation (46) may be modified so as to take 
account of this deviation, by writing v — b for v. The value of b depends upon the 
amount and nature of the gas, and its value for one gram of a gas is called the molec- 
ular volume of the gas. 

(b) The effect of mutual attraction of particles is to slow down the particles as they 
come into the layers of the gas adjacent to the walls. The attraction of the walls 
is constant and need not be considered. This slowing down of the particles makes 
the pressure of the gas less than it would otherwise be, by an amount which can be 
shown to be proportional to the square of the density of the gas or inversely pro- 
portional to the square of its volume. Equation (46) may be modified, so as to 
take account of this deviation, by writing p-\-a/v 2 for p. The quantity a is a con- 
stant for a given amount of a given gas. Equation (4b) therefore becomes 

(p + %){v-b)=MRT (13) 

39. Linde's liquid air machine. At very low temperatures and at very high pres- 
sures the cooling effect of free expansion is very considerable and it is utilized in 
Linde's liquid air machine. This machine operates as follows: Air under great pres- 
sure (150 to 200 atmospheres) is forced through a large coil of small copper tube at the 
end of which it escapes through a fine orifice into a low pressure region, whence it 
flows back over the coil of copper tube. The expansion of the air at the fine orifice 
cools the air slightly; this slightly cooled air in flowing back over the coil of copper 
pipe cools the inflowing air, which in its turn is further cooled when it passes through 

*A very complete discussion of gases, vapors, and liquids based upon van der 
Waals' equation is given in Nernst's Theoretical Chemistry. 



THE ATOMIC THEORY OF GASES. 349 

the orifice; the inflowing air is then still further cooled in the coil of pipe, and so on, 
until the temperature at the orifice is so greatly reduced that a portion of the air con- 
denses into a liquid which collects in the low-pressure chamber and is drawn off at 
will. If a liquid air machine were so arranged as to expand the high pressure air 
against a moving piston instead of allowing it to flow through an orifice, the doing 
of external work on the piston by the expanding gas would increase the cooling 
effect of the expansion and the efficiency of the liquid air machine would be increased. 
40. The perfect gas. A beginner may obtain a good idea of the thermal prop- 
erties of gases, sufficiently exact for all ordinary purposes, by assuming that Boyle's 
Law, Gay Lussac's Law, Joule and Thomson's principle, etc., are exactly true. All 
gases approach the ideal perfect gas at ordinary temperatures when highly rarefied, 
and the simple gases, that is to say, the gases of which the molecule is not complex, 
approximate very closely to the ideal perfect gas at ordinary temperatures and 
pressures. All gases depart widely from the ideal perfect gas at high pressures and 
low temperatures. 

Problems. 

48. Find the volume of 3.5 pounds of oxygen at a pressure of 3 atmospheres 
and at a temperature of 27°C, the volume of one pound of oxygen at o°C. and one 
atmosphere being 11.204 cubic feet. Ans. 14.36 cubic feet. 

49. One volume of oxygen, 2 atoms in the molecule, combines with 2 volumes 
of hydrogen, 2 atoms in the molecule, to form H 2 0. How many volumes of H 2 
vapor are produced? Ans. 2. 

50. One volume of chlorine, 2 atoms in the molecule, combines with one volume 
of hydrogen, 2 atoms in the molecule, to form HC1. How many volumes of HC1 
gas are formed? Ans. 2. 

51. Find the density in grams per cubic centimeter of a mixture of equal volumes 
of oxygen and hydrogen, the pressure of the mixture being 760 millimeters and the 
temperature of the mixture being o°C, the density of oxygen at o°C. and 760 mil- 
limeters being 0.00143 and the density of hydrogen at o°C. and 760 millimeters 
being 0.0000S9 grams per cubic centimeter. Ans. 0.0007595. 

Note. Each gas may be thought of as occupying the entire space alone at the 
given temperature and 380 millimeters pressure, and the density of each gas under 
these conditions calculated. The density of the mixed gas is then the sum of the 
densities of the two component gases. 

52. Calculate the square root of the average square of the velocities of hydro- 
gen molecules at o°C; and of oxygen molecules at o°C. 

Ans. Velocity of hydrogen molecules at o°C. 184,200 centimeters per second. 
Velocity of oxygen molecules at o°C. 46,050 centimeters per second. 

Note. The product nm in equation (11) page 345 is the density of the gas, so 
that if the density and pressure of the gas are known the value of « may be calcu- 
lated. In fact one needs only to know the ratio of pressure divided by density, and 
this depends only on the temperature. At 760 millimeters pressure ( = 1,013,000 
dynes per square centimeter) the density of hydrogen at o°C. is 0.00008954 gram 
per cubic centimeter and the density of oxygen is 16 times as great. 



CHAPTER V. 

THE SECOND LAW OF THERMODYNAMICS * 

41. A great deal of simple everyday knowledge is always 
taken for granted in a treatise on thermodynamics. In a pre- 
vious article it was stated that the important things in connection 
with the generation of steam in a boiler by the burning of coal 
are: (a) the temperature of the feed water, -(b) the temperature 
and pressure of the steam which is produced, (c) the character 
of the coal, (b) the temperature and composition of the air, and 
(e) the temperautre and composition of the flue gases. In a cer- 
tain sense this is true, but, of course, the fundamentally important 
thing is the knowledge that coal will burn and convert water 
into steam. Such fundamental knowledge is, however, always 
taken for granted in the study of thermodynamics. The nature 
of fire is not an object of study in thermodynamics, but every 
one knows what fire is in a simple practical way; every one 
knows that an object bscomes hot when it is placed on a hot 
stove; and every 'one knows that steam will squirt out of a hole 
in a steam boiler. 

42. The subdivisions of physical science. The science of 
mechanics applies to the more or less ideal phenomena which are 
associated with the motion of rigid bodies either singly or in 
connected machines; with regular motion of distortion of elastic 
bodies like the bending of a bow or the oscillation of a string; 
and with ideally simply motion of flow of liquids and gases, f 
In every actual case of motion, however, we always encounter 
turbulence more or less marked, and the science of mechanics, 

*Let the student remember that the term thermodynamics includes the whole 
of the theory of heat except the part which is based upon the atomic theory. Thus 
the preceding chapters, with the exception of Chapter IV, are chapters in ther- 
modynamics. 

fSee Arts. 85 and 122 of Mechanics. 

350 



THE SECOND LAW OF THERMODYNAMICS. 35 1 

which is the science of describing the phenomena of motion, 
fails completely if we attempt to consider the minute details 
which are involved in this turbulence. Thus it is fairly easy to 
understand how the structural parts of a bridge stretch and 
shorten as a train passes across the bridge if one does not attempt 
to take account of the extremely complicated effects due to ir- 
regular gusts of wind, and to the swaying and rattling of the 
cars. Or consider the movement of the water at a certain point 
in a brook ; there is indeed a fairly steady average velocity of the 
water at the point and a certain mean rhythmic variation, but 
superposed upon this average motion there is an erratic variation 
of velocity which is infinitely complicated. 

Fire is the most familiar example of a turbulent phenomenon, 
and its most striking characteristic is that its progress is not de- 
pendent upon any external driving cause; when once started 
it goes forward of itself, and with a rush. Tyndall,* in referring 
to this matter says that to account for the propagation of fire 
was one of the philosophical difficulties of the eighteenth century. 
A spark was found sufficient to initiate a conflagration, and the 
philosophical difficulty lay in the fact that the effect seemed to 
be beyond all proportion greater than the cause. In discussing 
this matter Tyndall refers to Boscovich's explanation of the 
sweeping character of fire. He pictures a high mountain rising 
out of the sea with sides so steep that blocks of stone are just 
able to rest upon them without rolling down. He supposes such 
blocks, diminishing gradually in size, to be distributed over the 
mountain, large blocks below, moderate sized blocks at the middle 
height, and dwindling to grains of sand at the top. A small 
bird touches with its foot a grain at the summit; it moves, sets 
the next larger grains in motion, these again let loose the pebbles, 
these the larger stones, and these the blocks; until finally the 
whole mountain-side rolls violently into the sea. 

The simple idea of cause and effect is a legitimate idea in the 
science of mechanics. Thus there is practically a definite rela- 

*See Tyndall's Heat a Mode of Motion, page 66. 



35 2 THEORY OF HEAT. 

tionship between the amount of load on a bridge and the extent 
to which the bridge sags ; but the simple idea of cause and effect 
cannot be applied to physical phenomena which involve turbu- 
lence. For example, the sun's rays heat the air next the ground 
over a large stretch of country, thus producing an unstable state 
of the atmosphere. Under these conditions, an extremely slight 
disturbance at one point may start the warm air moving upward, 
and from these slight beginnings a more and more violent chim- 
ney-like effect may develop, and lead to one of those great atmo- 
spheric movements which are called cyclones; and whether the 
cyclonic movement brings a severe storm to one or another part 
of the country may depend upon some insignificant character of 
the original infinitesimal disturbance, such as its place or time of 
occurrence.* 

Every physical phenomenon involving turbulence is to some 
extent self-sustaining, every such phenomenon has a certain 
impetuous quality, and these remarkable characteristics of tur- 
bulence are now definitely formulated as the second law of 
thermodynamics. 

The most important practical thing in connection with the 
turbulent aspect of any physical phenomenon is its general result 
or consequence, just as the important thing about the burning 
of a house is the loss. How utterly useless and uninteresting it 
would be, for example, to study the minutest details of a con- 
flagration (assuming such study to be possible), recording the 
height and breadth and the irregular and evanescent distribution 
of temperature throughout each flicker of consuming flame, the 
story of each crackling sound and of every yield and sway of 
timber and wall! The fact is, we are immersed in an il- 
limitable sea of phenomena every single detail of which is infi- 
nitely manifold, and no completely adequate science can ever 
be developed. 

Physical science, aside from those branches which are depend- 

*See a very brief article by W. S. Franklin in Science, Volume 14, pages 496- 
497, September 27, 1901. 



THE SECOND LAW OF THERMODYNAMICS. 353 

ent upon the atomic theory, consists of three branches, namely: 
(i) Mechanics, including Hydraulics, Electricity and Magnetism, 
Light and Sound ; the science of those phenomena in which tur- 
bulence may for practical purposes be ignored; (2) Statistical 
Physics, the science of those phenomena in which turbulence 
introduces an appreciable and practically important erratic ele- 
ment. Such phenomena can be studied only by the statistical 
method (the record of individual cases and the study of averages). 
Meteorology is the best example of statistical physics,* although 
every physical phenomenon has its statistical aspect; and (3) 
Thermodynamics. Some of the features of thermodynamics have 
already been pointed out. It is the study of changes of state of 
substances. A most important aspect of thermodynamics re- 
mains, however, to be considered, and a preliminary idea of this 
new aspect may be obtained by means of an analogy. In every- 
day life we see the fire-insurance companies concerned with cer- 
tain broad features of statistical physics in their examinations 
and records of fires, and we see them also concerned with a profit 
and loss account which is wholly abstracted from the details of 
the phenomena of conflagration. Thermodynamics is the profit 
and loss branch of physics as it were ; and like the profit and loss 
branch of fire insurance, thermodynamics is completely abstracted 
from any consideration of the details of any physical phenomenon. 
Thermodynamics is concerned with the measurement and account- 
ting of that type of physical degeneration which accompanies 
turbulence just as fire insurance is concerned with the estimation 
and accounting of what we might call, using a fine phrase, struc- 
tural degeneration by fire. 

Thermodynamic degeneration. Every one has a feeling of the 
irreparable effects of disaster. The collapse of a bridge, the de- 
struction of a house by fire or the wreck of a ship involve 
loss which indeed may be forgotten after reconstruction, 
but never balanced. The havoc that is wrought is essentially 

*See two brief articles by W. S. Franklin, Transactions of American Institute 
of Electrical Engineers, Vol. 20, pages 285-286; and Science, Vol. 14, pages 496-497, 
September 27, 1901. 
24 



354 THEORY OF HEAT. 

irreparable. It is desirable to use the word degeneration in a 
very narrow technical sense when we come to consider the second 
law of thermodynamics, and the way may be paved to a clear 
understanding of the later and accepted use of this word in 
physics by applying it now to designate that aspect of disaster 
which is irreparable. The burning of a building, for example, is 
a process of degeneration. The term thermodynamic degeneration 
applies to the effects of turbulence. Thus, a certain degeneration 
is associated with the turbulence which is produced when a hot 
iron is dipped into water, a certain degeneration is associated with 
the escape of a compressed gas through ah orifice, a certain de_ 
generation is associated with the flow of heat from a region of 
high temperature to a region of low temperature, a certain de- 
generation is associated with the conversion of work into heat 
by the rubbing of a coin on a board, and so on. 

43. Reversible processes. A substance in thermal equilib- 
rium is generally under the influence of external agencies. Thus 
surrounding substances confine a given substance to a certain 
region of space, and they exert upon the given substance a definite 
constant pressure; surrounding substances are at the same tem- 
perature as the given substance, and, according to the atomic 
theory, the molecules of the given substance rebound from sur- 
rounding substances with their motion on the average unchanged ; 
surrounding substances may exert constant magnetic or electric 
influences upon the given substance; and so on. If the external 
influences which act upon a fluid* in thermal equilibrium are made 
to change very slowly causing the pressure, volume, and temperature 
of the fluid to pass very slowly through a continuous series of values, 
and in general involving the doing of work upon or by the fluid, 
and the giving of heat to or taking of heat from the fluid, then the 
fluid will pass slowly through a process consisting of a continuous 
series of states of thermal equilibrium. Such a process is called 

*The thermodynamics of solids is extremely complicated except in a very few 
particulars. Therefore most of the general statements in the present discussion 
are limited to liquids and gases. 



THE SECOND LAW OF THERMODYNAMICS. 355 

a reversible process, for the reason that the fluid will pass through 
the same series of states in reverse order if the external influences 
are changed slowly in a reversed sense. The characteristics of a 
reversible process are therefore as follows: 

(a) A substance which undergoes a reversible process must be 
under varying external influences. A closed system* cannot per- 
form a reversible process. 

(b) A substance as it undergoes a reversible process is at each 
instant in a state of thermal equilibrium. If, at a given instant 
during a reversible process, the external influences should cease 
to change, no commotion would be left in the substance, and it 
would be on the instant in thermal equilibrium. 

(c) A reversible process must take place slowly, indeed with 
infinite slowness. An actual process, that is a process which 
actually does proceed, can only be approximately reversible. 

Examples of reversible processes. The very slow compression 
or expansion of a gas in a cylinder by the motion of a piston is a 
reversible process. The very slow heating of a gas in a closed 
vessel by placing the vessel in contact with a very slightly warmer 
substance is a reversible process because the gas is at each instant 
sensibly in thermal equilibrium. 

44. Irreversible processes or sweeps. When a substance is 
settling or tending to settle to thermal equilibrium it may be 
said to undergo a process. Such a process cannot be arrested 
and held at any stage short of complete thermal equilibrium, but 
it always and inevitably proceeds towards that state. Such a 
process may, therefore, be called a sweeping process or simply a 
sweep. Consider, for example, the action which takes place 
when a red hot piece of iron is dropped into a pail of water. As 
the entire system settles to thermal equilibrium it passes through 
a series of stages each one of which grows out of the preceding 
stage inevitably, and it is impossible to arrest the process at any 
stage short of complete thermal equilibrium. 

Molecular conception of a sweeping process. The molecular 

*See Mechanics, Art. 58. 



356 THEORY OF HEAT. 

theory enables one to form a mental picture of a sweeping process. 
Thus Boscovich's idea of what we nowadays call an irreversible 
process or sweep is given in Art. 42 and it would be scarcely 
possible to improve on Boscovich's description in so far as the 
two most important characteristics of a sweeping process are 
concerned, namely, that a sweep is not dependent upon an exter- 
nal driving cause and that a sweep once started proceeds inevit- 
ably to a definite end. 

45. Simple sweeps. The settling of a closed system to thermal 
equilibrium may be called a simple sweep. For example, the 
equilibrium of a mixture of oxygen and hydrogen in a closed 
vessel may be distributed by a minute spark, and the explosion 
of the gases together with the subsequent settling of the water 
vapor to a quiescent state constitutes a simple sweep. The equi- 
librium of a gas confined under high pressure in one half of a two- 
chambered vessel may be disturbed by opening a cock which 
connects the two chambers, and the rush of gas into the empty 
chamber constitutes a simple sweep. 

46. Trailing sweeps. When external influences change con- 
tinuously a substance in its tendency to settle to thermal equi- 
librium never catches up as it were with the changing conditions, 
but trails along behind them, and we have what may be called a 
trailing sweep. Thus, the rapid expansion or compression* of a 
gas in a cylinder is a trailing sweep. So long as the piston 
moves at a perceptible speed the gas in its tendency to settle 
to equilibrium never catches up with the varying conditions. 
This is evident when one considers that a sudden stopping of 
the piston would leave some slight turbulence in the gas which 
would not be the case if the gas were in equilibrium at the instant 

*The different effects here mentioned in connection with rapid versus slow ex- 
pansion must not be confused with the distinction between isothermic expansion 
and adiabatic expansion of gases. See Arts. 56 and 57. Isothermic expansion 
or compression must be slow in order that the necessary heat may be given to or 
taken from the gas to keep its temperature constant. On the other hand, any ap- 
proximation to adiabatic expansion or compression of a gas in a cylinder must be 
rapid in order that there may be no appreciable amount of heat given to or taken 
from the gas by the cylinder walls. 



THE SECOND LAW OF THERMODYNAMICS. 357 

the piston stopped. When the piston is moved more and more 
slowly, however, the departure of the gas from strict thermal 
equilibrium at each stage of the expansion or compression be- 
comes less and less, and the expansion or compression approaches 
more and more nearly to a reversible process. 

The rapid heating or cooling of a gas in a closed vessel is a 
trailing sweep. So long as heat is given to or taken from the 
gas at a perceptible rate there will be perceptible differences of 
temperature in different parts of the gas; and the gas in its ten- 
dency to settle to thermal equilibrium never catches up with the 
increasing or decreasing temperature of the walls of the contain- 
ing vessel. When the gas is heated or cooled more and more 
slowly, that is, when heat is given to or taken from the gas at a 
rate which becomes more and more nearly imperceptible, then 
the departure of the gas from strict thermal equilibrium at each 
stage of the heating or cooling process becomes less and less and 
the heating or cooling process approaches more and more nearly 
to a reversible process. 

47. Steady sweeps. A substance may be subjected to external 
action which although permanent or unvarying is incompatible 
with thermal equilibrium. When such is the case the substance 
settles to a permanent or unvarying state which is not a state of 
thermal equilibrium. Such a state of a substance may be called 
a steady sweep. For example, the two faces of a slab or the two 
ends of a wire may be kept permanently at different temperatures, 
and when this is done, the slab or wire settles to an unvarying 
state which is by no means a state of thermal equilibrium. Heat 
flows through the slab or along the wire from the region of high 
temperature to the region of low temperature, never from the 
region of low temperature to the region of high temperature. 
This flow of heat through the slab, or along the wire, is an irre- 
versible process and it constitutes a steady sweep. The ends of 
a wire may be connected to a battery or dynamo so that a con- 
stant electric current flows through the wire, and the heat which 
is generated in the wire by the current may be steadily carried 



35 8 THEORY OF HEAT. 

away by a stream of water or air. Under these conditions the 
wire settles to an unvarying state which is by no means a state 
of thermal equilibrium, the battery or dynamo does work on the 
wire, and this work reappears steadily as heat in the wire. A 
reversal of the current does not reverse this process and cause 
heat energy to disappear in the wire (cooling of the wire) and 
reappear as work done in driving the dynamo as a motor or in 
recharging the battery; the process is irreversible and it con- 
stitutes a steady sweep. 

48. Thermodynamic degeneration. Every one must admit 
that the impetuous character of a sweeping process suggests a 
certain havoc, a certain degeneration in the substance or system 
in which the sweep takes place. Consider, for example, a charge 
of gunpowder which has been exploded; if it is exploded in a 
large empty vessel, everything is there after the explosion, all 
of the energy is there and all of the material substance is there, 
but it cannot be exploded a second time.* 

At this point of our discussion it is necessary to use the word 
degeneration so as to express more or less tentatively the idea 
that every sweeping process brings about a definite amount of 
degeneration, an amount that can be expressed numerically just 
as one speaks of so many pounds of sugar or so many yards of 
cloth. Thus, a certain amount of degeneration is brought about 
when a compressed gas escapes through an orifice, a certain 
amount of degeneration is brought about when heat flows from 
a region of high temperature to a region of low temperature, 
a certain amount of degeneration is brought about when work is 
converted into heat by friction or by the flow of an electric current 
through a wire, and so on. 

*The man on the street has heard much during recent years of the conservation 
of energy and of the conservation of matter, and the old proverb that "you can't 
eat your cake and have it" presents to his mind a very simple fact concerning use 
and waste which in its less familiar aspects, as relating to engines for example, he 
tries in vain to rationalize in terms of these principles of conservation ! Nearly all 
of the intuitive sense of the man on the street concerning use and waste (and he 
has a great deal) is involved in the second law of thermodynamics which is not a 
law of conservation at all. It is a law of waste. 



THE SECOND LAW OF THERMODYNAMICS. 359 

In a simple sweep the degeneration lies wholly in the relation 
between initial and final states.* This is necessarily the case 
because no outside substance is affected in any way by the sweep, 
no work is done on or by the substance which undergoes the 
sweep and no heat is given to or taken from it. In a trailing 
sweep the degeneration may lie partly in the relation between 
the initial and final states of the substance which undergoes the 
sweep, partly in the conversion of work into heat and partly in 
the flow of heat from a high temperature region to a low tempera- 
ture region. In a steady sweep, however, the substance which 
undergoes the sweep remains entirely unchanged as the sweep 
progresses, and the degeneration lies wholly in the conversion of 
work into heat, in the transfer of heat from a region of high tempera- 
ture to a region of low temperature, or in both. Therefore the idea 
of thermodynamic degeneration as a measurable quantity can 
be reached in the simplest possible manner by a careful scrutiny 
of a steady sweep. 

Proposition (a). The thermodynamic degeneration which is rep- 
resented by the direct conversion of work into heat at a given tem- 
perature is proportional to the quantity of work so converted. 
Consider, for example, a steady flow of electric current through a 
wire from which the heat is abstracted so that the temperature 
remains constant. This process is steady, that is to say, it re- 
mains unchanged during successive intervals of time, and there- 
fore any result of the process must be proportional to the time 
which elapses, that is to say, the amount of degeneration occur- 
ring in a given interval of time is proportional to the time, but 
the amount of work which is degenerated into heat is also pro- 
portional to the time. Therefore the amount of degeneration is 
proportional to the amount of work converted into heat at the 
given temperature. 

Proposition (b). The thermodynamic degeneration which is 
represented by the transfer of heat from a given high temperature T x 

*The possibility of assigning a definite amount of thermodynamic degeneration 
to a given change of state of a substance depends upon Clausius's theorem which 
is discussed in Art. 54. 



360 THEORY OF HEAT. 

to a given low temperature T 2 is proportional to the quantity of heat 
transferred. Consider a steady flow of heat from temperature T x 
to temperature T 2 constituting a steady sweep, a sweep which 
remains entirely unchanged in character in successive intervals 
of time. Any result of this sweep must be proportional to the 
lapse of time, and therefore the degeneration which takes place 
in a given interval of time is proportional to the time; but the 
quantity of heat transferred is also proportional to the time, 
therefore the amount of degeneration is proportional to the 
quantity of heat transferred from temperature T x to temperature 
T 2 . 

Definition of the ratio of two temperatures. The definition of 
the ratio of two temperatures which is given in Art. 6 was under- 
stood to be a provisional definition. We are now in a position 
to propose a definition of the ratio of two temperatures which is 
independent of the physical properties of any particular sub- 
stance. This definition will remain somewhat vague, however, 
until the action of the steam engine is discussed in the next article. 
According to proposition (a) above, the thermodynamic degenera- 
tion which is involved in the conversion of work into heat at a 
given temperature is proportional to the amount of work so 
converted, and the proportionality factor depends upon the tem- 
perature only. Therefore we may write 

<p' = m y W (i) 

J = m.JV (ii) 

where <p' is the degeneration involved in the conversion of an 
amount of work W into heat at temperature T lt ip" is the de- 
generation involved in the conversion of an amount of work W 
into heat at temperature T 2 , and m l and m 2 are factors which 
depend upon 7\ and T 2 respectively. 

The amount of work W having been converted into heat at 
temperature 7\, imagine the heat to flow to a lower temperature 
T 2 , thus involving an additional amount of degeneration accord- 



THE SECOND LAW OF THERMODYNAMICS. 361 

ing to proposition (b) above. The conversion of work W into 
heat at temperature 7\ and the subsequent flow of this heat to a 
lower temperature T 2 gives the same result as would be produced 
by the conversion of the work into heat at the lower temperature 
directly. Therefore, the lower the temperature at which work 
is converted into heat the greater the amount of degeneration 
involved. That is to say, the factor m 2 in equation (ii) is larger 
in value than the factor m l in equation (i), temperature T Y being 
higher* than temperature T 2 . Therefore, since m 1 and m 2 depend 
only upon 7\ and T 2 respectively, it is permissible to adopt the 
equation 

T x m 2 



T 2 



(iii) 



as the definition of the ratio Tj T 2 . This definition of tempera- 
ture ratios will be explained more in detail in the next article. 
The definition is due originally to Lord Kelvin. 

Another way to express the definition which is involved in 
equation (iii) is to consider that the factor m l is the smaller the 
higher the temperature T x , so that we may adopt k/m l as the 
measure of the temperature T v and kfm 2 as the measure of the 
temperature T 2 , giving 

«! = y ( lv ) 

and 

m 2 = Y (v) 

1 2 

where k is an indeterminate constant. Therefore equation (i) 
and (ii) may be written in the general form 

kW 

*The idea of higher and lower temperature is not dependent upon any method 
of measuring temperature. When a substance receives heat definite observable 
effects are produced, and when these effects are produced by placing one substance 
into contact with another substance, the other substance is known to give heat to 
the given substance and its temperature is known to be higher than the temperature 



362 THEORY OF HEAT. 

where <p is the thermodynamic degeneration involved in the con- 
version of an amount of work W into heat at temperature T, and 
k is an indeterminate constant. 

The ratio of two temperatures as defined by equation {Hi) is 
very nearly the same as the ratio of two temperatures as measured 
by the gas thermometer, and therefore gas thermometer temperatures 
(so called absolute temperatures as measured by the gas thermometer) 
may be used throughout this chapter without appreciable error.* 
Now, since the factor k in equation (vi) is indeterminate we may 
choose as our unit of thermodynamic degeneration the amount 
which is involved in the conversion of one unit of work into heat 
at a temperature of one degree on the "absolute" scale; then the 
value of k is unity and equation (vi) becomes 

W ( ^ 

V = -f ( J 4) 

When W is expressed in joules and T in degrees centigrade, <p is 
expressed in terms of joules per degree. Thus one joule per 
degree is the degeneration involved in the conversion of one 
joule of work into heat at i°C. absolute, or the amount involved 
in the conversion of 1000 joules into heat at iooo°C. absolute. 
To convert an amount of work W into heat at temperature T x 
involves W\T X units of degeneration, to convert the same 
amount of work into heat at temperature T 2 involves W/T 2 

of the given substance. Thus a piece of wax is melted when it is placed on a hot 
stove. 

*If the gas used in the gas thermometer conforms exactly to Boyle's Law and 
if the temperature of the gas would neither rise nor fall during free expansion, then 
absolute temperatures as measured by the gas thermometer would coincide exactly 
with temperatures as defined by equations (iv) and (v). This proposition is es- 
tablished in Art. 58 on the basis of the theory of the perfect engine as outlined in 
Art. 50. 

If the gas used in the gas thermometer does not conform to Boyle's Law (and 
no gas does conform to Boyle's Law rigorously) and if the gas does change its tem- 
perature during free expansion (and all gases do more or less), then temperatures 
as measured by the gas thermometer depart slightly from temperatures as defined 
by equations (iv) and (v). This matter is discussed fully in Buckingham's Ther- 
modynamics, pages 127-136 (The Macmi-llan Company, New York, 1900). 



THE SECOND LAW OF THERMODYNAMICS. 363 

units of degeneration, and therefore to transfer an amount of 
heat equal to W from temperature T L to temperature T 2 must 
involve an amount of degeneration equal to the excess of W j T 2 
over Wj 7\ or an amount equal to 



W 



(} r ~) or ^(f-^J 



where H is the amount of heat transferred. 

49. The second law of thermodynamics, (a) The thermo- 
dynamic degeneration which accompanies an irreversible process 
cannot be directly repaired, nor can it be repaired by any means 
without compensation. This is an entirely general statement of 
the second law of thermodynamics. The direct repair of the 
degeneration due to a sweeping process means* the undoing of 
the havoc wrought by the process by allowing the sweeping 
process to perform itself backwards, an idea which is exactly as 
absurd as the idea of allowing a burned house to unburn itself. 
Following are several specialized statements of the second law 
of thermodynamics. 

(b) Heat cannot pass directly from a cold body to a hot body, nor 
can heat be transferred from a cold body to a hot body by any means 
without compensation. 

(c) Heat cannot be converted directly into work, nor can heat be 
converted into work by any means without compensation. 

The direct conversion of heat into work would be the simple 
reverse of any of the ordinary sweeping processes which involve 
the degeneration of work into heat (direct conversion of heat 
into work would be to allow the sweeping process to perform itself 
backwards). For example, work is degenerated into heat in the 
bearing of a rotating shaft, and we all know that to reverse the 
motion of the shaft does not cause the bearing to grow cold and 
the heat so lost to appear as work helping to drive the shaft. 
That would be a rotary engine indeed! There is an important 
general theorem in thermodynamics to the effect that if two 
sweeping processes A and B involve the same amount of de- 



364 THEORY OF HEAT. 

generation, and if either of the processes, say A, has been allowed 
to perform its sweep, then by a lever arrangement, as it were, 
the process B can be carefully let down, and the havoc wrought 
by the sweep of A can be undone. The result of this operation, 
however, would be to leave the system B in the condition into 
which it would be degenerated if the process B had been allowed 
to sweep instead of being let down. This is very much as if, 
having two similar houses A and B, one of which A has been 
burned, we could rig up a mechanism which would let down B to 
ashes and cause A to be restored in the original actual materials 
of which it was first constructed. This is of course impossible 
in the case of the two houses but it is possible in every known 
case of thermodynamic degeneration. This general theorem is 
as thoroughly established as any generalization in physics, and 
if it is true and if we ever find a way to convert heat into work 
unconditionally and without cost or compensation, then it will 
be proven indirectly that a shaft can be driven by heating one of 
the bearings in which it rotates, for direct conversion of heat into 
work by one process must be according to this general theorem 
equivalent to and replaceable by the reverse of any ordinary 
sweeping process which converts work into heat. 

(d) A gas cannot pass directly from a region of low pressure to a 
region of high pressure, nor can a gas be transferred from a region 
of low pressure to a region of high pressure by any means without 
compensation. 

Imagine a gas squirting itself backwards through a nozzle into 
a high pressure reservoir ! The repeated statement of self-evident 
facts concerning direct repair in these statements of the second 
law of thermodynamics may seem ridiculous to the intelligent 
reader, but the second law of thermodynamics is a statement of a 
fact which every one knows coupled with a generalizing clause 
which once thoroughly understood is almost if not quite self- 
evident.* 

*Here is one more statement of the second law of thermodynamics, the oldest 
English version of it: 



THE SECOND LAW OF THERMODYNAMICS. 365 

Let us return to the fourth statement (d) and consider with the 
help of an example what is meant by compensation in its thermo- 
dynamic sense. A gas can be transferred from a region of low 
pressure to a region of high pressure by means of a pump, and 
the work that is done in driving the pump, even supposing the 
pump to be frictionless, is all converted into heat. This con- 
version of work into heat is the necessary cost or compensation 
for the transfer of the gas from a low pressure region to a high 
pressure region. 

Consider the second statement (b) ; in an artificial-ice factory 
heat is continually abstracted from the freezing room and trans- 
ferred to the warm outside air; but to accomplish this result, 
even by an ideally perfect, frictionless mechanism, a certain 
amount of work is required to drive the ammonia pump and this 
work is converted into heat (the amount of heat that is delivered 
to the warm air outside exceeds the heat that is abstracted from 
the cold room). This conversion of work into heat compensates 

Humpty Dumpty sat on a wall, 

Humpty Dumpty had a great fall, 

All the king's horses and all the king's men, 

Cannot put Humpty Dumpty together again. 

This is perhaps the most sensible of all the statements of the second law, for which 
we will allow it to pass for the moment, inasmuch as it ignores direct repair and 
refers at once to the most powerful of external means. It is important, however, to 
remember that in Humpty Dumpty's case we are concerned with structural de- 
generation, not with the much simpler kind of degeneration in a structureless fluid 
due to turbulence. 

The second law of thermodynamics, of all the generalizations of physics, is cer- 
tainly the most deeply seated in the common sense of all men, and one of the most 
humorous of children's verses refers to the man whose wondrous wisdom enabled 
him to circumvent it by "direct repair" : 

There was a man in our town 
And he was wondrous wise 
He jumped into a bramble bush 
And scratched out both his eyes. 
And when he found his eyes were out 
With all his might and main 
He jumped into another bush 
And scratched them in again. 



Z66 



THEORY OF HEAT. 



for the transfer of heat from the freezing room to the warm region 
outside. 

Consider the third statement (c) ; in an ordinary steam engine 
heat is converted into work, but to accomplish this transforma- 
tion a large quantity of heat must be supplied to the engine at 
high temperature and some of this heat (about nine-tenths of it 
in even the very best of steam engines) must be let down, as it 
were, to the low temperature of the exhaust to compensate for 
the conversion of the remainder into work. 

50. Engines and refrigerating machines. An engine, or to be 
more specific, a heat engine is a machine for converting heat into 



f,IJ}k condensing 

"^•AV\ water 




fire 



water 



water 



Fig. 26. 



mechanical work. The essential organs of an engine are shown 
in Fig. 26. A certain amount of heat is supplied to the engine 
with the steam, a portion of this heat is transformed into work, 
and the remainder is delivered with the exhaust steam to the 
condenser. 

A refrigerating machine is a machine for utilizing mechanical 
work for the extraction of heat from a cold region and its delivery 



THE SECOND LAW OF THERMODYNAMICS. 367 

to a hot region. The essential organs of a refrigerating machine 
are shown in Fig. 1 7. The work used in driving the pump which 
is shown in Fig. 17 is converted into heat, and this heat, together 
with the heat which is extracted from the cold region, is delivered 
to the hot region. 

The ideal or perfect engine. A most important theorem con- 
cerning the possible efficiency* of a heat engine may be estab- 
lished by means of an argument based on the assumption that 
the operation of the engine involves no irreversible or sweeping 
processes of any kind. The ordinary steam engine does involve 
such processes; the engine is subject to friction and the steam as 
it passes through the engine undergoes a variety of sweeping 
processes;! but if the engine were frictionless, if it were driven 
slowly, if the cylinder were prevented from cooling the steam, 
and if the steam were expanded sufficiently to prevent puffing, 
then the processes involved in the operation of the engine would 
be reversible. Such an ideal engine is called a reversible engine 
or a perfect engine. % 

During a given time the engine which is represented in Fig. 26 
takes a certain quantity of heat H 1 from the boiler at temperature 
T lf develops a certain amount of work W (in excess of the work 
required to drive the feed-water pump) and delivers the remainder 
of the heatH 2 to the condenser at temperature T 2 . 

According to the first law of thermodynamics, the work W 
must be equal to (H x — H 2 ), H l and H 2 being both expressed 
in energy units. Therefore 

W = H x - H 2 (i) 

*The efficiency of an engine is the fraction of the supplied heat which is con- 
verted into work by the engine. 

tSee Art. 51. 

Jin order that the entire action which takes place in Fig. 26 may be reversible, 
the feed-water pump must be arranged to heat the water which it takes from the 
condenser from condenser temperature T 2 up to boiler temperature T x . This action 
of the feed-water pump could be realized by arranging for it to take not only water 
from the condenser but also a certain amount of uncondensed steam. The com- 
pression of this mixture of steam and water under the piston of the feed-water pump 
would condense the steam, and the condensation would heat the feed water to 
boiler temperature. 



368 THEORY OF HEAT. 

For the sake of simplicity of argument the net result of the 
operation of the engine may be thought of as (a) The conversion 
into work the whole of the heat H x which is taken from the boiler 
at temperature 7\, and (b) The reconversion of the portion H 2 
back into heat at temperature T 2 . The regeneration* associated 
with process (a) is equal to H l / T x according to equation (14), and 
the degeneration associated with process (b) is equal to H 2 / T 2 
according to equation (14). If the operation of the engine in- 
volves no sweeping processes, that is to say, if the engine is a 
reversible or perfect engine, then there can be, on the whole, no 
degeneration associated with the operation of the engine so that 
the regeneration which is associated with the process (a) must 
be balanced by the degeneration which is associated with process 
(&), or, in other words, we must have 





H 2 

T- 2 


H 2 


T, 



or 

(iS) 

This equation is the one upon which Lord Kelvin based his 
thermodynamic definition of the ratio of two temperatures. The 
meaning of the equation is as follows: Given a perfect engine 
working between boiler temperature T x and condenser tempera- 
ture T 2 ; and let H l be the amount of heat taken by the engine 
from the boiler and H 2 the amount of heat delivered by the engine 
to the condenser in a given time. Then the ratio of H l to H l 
is equal to the ratio of T x to T 2 . 

Efficiency of a perfect engine. Substituting the value of H 2 
from equation (i) inequation (15), and solving for W, we have 

T — T 
W = ±= 1 ---H l (16) 

1 1 

*To convert an amount of work W into heat at a given temperature involves 
an amount of degeneration, and to convert the heat into work would involve the 
same amount of what may be called thermodynamic regeneration. 



THE SECOND LAW OF THERMODYNAMICS. 369 

The fractional part ( T x — T 2 ) / T x of the heat H x is converted into 
work by the engine, and this fraction is called the efficiency of the 
engine. It follows from equation (16) that all reversible engines 
have the same efficiency for given values of the temperatures T x and 
T 2 whatever the nature of the working fluid may be. 

Efficiency of an imperfect engine. If the operation of the engine 
involves sweeping processes of any kind, the degeneration H 2 / T 2 
which is associated with process (b) above mentioned must be 
greater than the regeneration H [ /T l which is associated with 
process (a) ; or, in other words 

Hi H 2 

or, substituting the value of H 2 from equation (i) and solving 

for W, we have 

T — T 

w < ^v^ • #i 

1 1 

Therefore, comparing this inequality with equation (16), it follows 
that any irreversible engine (an engine which involves sweeping 
processes of any kind in its operation) working between temperatures 
T x and T 2 has less efficiency than a reversible engine working between 
the same temperatures. 

The refrigerating machine.* Figure 27 is a diagram for fixing in the reader's 
mind the various temperatures and quantities of heat and work which are involved 
in the operation of the refrigerating machine. The machine is representing by 
MM. The chamber in which evaporation takes place (the boiler) is at low tem- 
perature Ti, and the chamber in which condensation takes place (the condenser) 
is at high temperature T v The result of the operation of the machine for a given 
time is the abstraction of a defininite quantity of heat Hi from the cold region, the 
conversion into heat of the definite amount of work W which has been used to drive 
the machine, and the delivery of an amount of heat H 1 to the hot region. 

According to the first law of thermodynamics, the heat H x which is delivered 
to the hot region is equal to Hi-\-W, or 

W=H X -H» (i) 

For the sake of argument the net result of the operation of the refrigerating 
machine may be thought of as (a) the conversion into work of all of the heat H% which 

♦See Art. 18. 

2 5 



370 



THEORY OF HEAT. 



/ w / 

/driving belt 




engine for recovering 
work from returning liquid 

Fig. 27. 

is abstracted from the cold region at temperature Ti, and (&) the reconversion into 
heat at temperature T y of the whole of this work (Hi) together with the work W 
that is expended in driving the pump. Process (a) involves an amount of regen- 
eration equal to HiJTi, and process (&) involves an amount of degeneration which 
is equal to H X \T X (where H 1 = Hi-\-W). If all the processes involved in the 
operation of the refrigerating machine were reversible, its operation would not in- 
volve on the whole any degeneration, and therefore we would have 



t 2 






or, substituting the value of H v from equation (i) and solving for Hi we have 



H 2 = 



r 2 



W 



(17) 



r 1 -r 2 

in which Hi is the amount of heat abstracted from the cold room by a perfect 
refrigerating machine during the expenditure of an amount of work W in driving 
the machine, Ti being the temperature of the region from which the heat is taken 
and T x the temperature of the region to which the heat is delivered. 

51. Conditions which limit the efficiency of engines in prac- 
tice.* A fraction of the heat which is delivered to an engine is 

*Friction between moving parts of an engine has nothing to do with the processes 
undergone by the steam as it passes through the engine, and therefore friction is 
not mentioned in the following discussion. Friction does, however, have an im- 
portant influence on efficiency. 



THE SECOND LAW OF THERMODYNAMICS. 37 1 

converted into work. In order that this fraction may be large, 
the ratio Tj T 2 must be as large as possible, and sweeping 
processes must be obviated as much as possible in the operation 
of the engine as explained in Art. 50; T x being the temperature 
of the steam supplied to the engine and T 2 the temperature of 
the exhaust steam. The ratio of the initial temperature to the 
final temperature of the expanding steam (or gas) in the engine 
depends upon the ratio of the initial volume to the final volume 
of the steam (or gas). 

In order that moderately small* cylinders may be used for the 
development of a given amount of power, the initial pressure of 
the steam or gas must be high; and in order that the final tem- 
perature may not be lower than atmospheric temperature or 
lower than available condenser-water temperature, the initial 
temperature of the steam or gas must be high. 

The boiler temperatures used in ordinary steam engine practice 
are from I70°C. (which corresponds to about 100 pounds per 
square inch above atmospheric pressure) to 200°C. (which cor- 
responds to about 200 pounds per square inch above atmospheric 
pressure), f The lowest feasible exhaust steam temperature is 
about 40°C. In the gas engine the initial temperature is the 
temperature of the mixture of air and gas immediately after the 
explosion which may be i7oo°C. or higher on the "absolute" 
scale, and the temperature is reduced by expansion to about 
half this value. The mixture of gas and air in a gas engine is 

*The objections to large cylinders are: (a) Their great cost, (b) the great amount 
of heat radiated by them, (c) the great amount of cylinder condensation in a large 
cylinder as explained later, (d) the great amount of piston friction and (e) cost of 
lubrication of piston. 

fA boiler explosion is especially disastrous when the boiler contains a large 
quantity of water, and therefore the only type of boiler which is considered safe 
for very large pressures (and high temperatures) is the so-called flash boiler which 
is used in some steam automobiles. This boiler consists of a coil of small iron pipe; 
feed water is pumped into such a boiler continuously and converted almost at once 
into steam upon its entrance into the boiler. The temperature of the steam de- 
livered by such a boiler is frequently as high as 2j$o°C. which corresponds to a 
pressure of about 500 pounds per square inch. 



37 2 THEORY OF HEAT. 

always compressed before the explosion in order to enable an en- 
gine with small-sized cylinders to develop a large amount of power. 
The sweeping processes which the steam in a steam engine 
undergoes are as follows: 

(a) Wire drawing. If the pipes and passages traversed by 
the steam from the boiler to the engine are small, the pressure in 
the cylinder with open ports will be lower than boiler pressure, 
so that the entering steam passes from a region of high pressure 
into a region of low pressure. Also as the cut-off valve closes, 
steam will rush into the cylinder through a narrowing aperture. 
This effect is called wire drawing, and to provide against loss of 
efficiency from this cause, the pipes must be of ample size and 
the cut-off valve must operate very quickly. 

(b) Radiation. The cooling of pipes and cylinder by the giving 
of heat to surrounding cooler bodies is a sweeping process, and is 
to be obviated as much as possible by covering pipes and cylinder 
with a thick coating of porous insulating material. 

(c) Cylinder condensation. As a charge of steam in the cylin- 
der expands it cools and cools the cylinder and piston, so that 
when steam is next admitted it heats the cylinder and piston up 
again and is itself cooled. This effect cannot be eliminated, but 
it can be largely reduced by providing separate passages for the 
ingress and egress of steam and by using a series of cylinders of 
increasing size, the smallest cylinder being arranged to take steam 
directly from the boiler and exhaust into the next larger cylinder 
which in turn exhausts into a still larger cylinder, and so on. 
In this way the range of temperature in each cylinder is small and 
the effects of cylinder condensation are greatly reduced. A steam 
engine in which expansion of the steam takes place in two stages 
(in two cylinders) is called a compound engine. A steam engine 
in which the expansion of the steam takes place in three stages 
(in three cylinders) is called a triple expansion engine. 

The loss of efficiency due to cylinder condensation is greatly 
reduced by the use of superheated steam because the exchange 
of heat between the steam and the cylinder walls is very greatly 



THE SECOND LAW OF THERMODYNAMICS. 373 

reduced when the steam does not condense. Thus S. LeRoy 
Brown has found that heat is imparted to a cool metal surface 
about forty times as fast by condensing steam as by a gas at the 
same temperature. 

(d) Effect of high piston velocity. If the piston speed is too 
great, the pressure of the expanding steam becomes ineffective 
because the portions of the steam near the moving piston are 
expanded and cooled before the more remote parts of the steam 
are affected. This effect is negligible at the highest piston ve- 
locities which are mechanically feasible. 

(e) Puffing. The steam at the end of a stroke is usually at a 
pressure which exceeds the pressure in the condenser and it 
rushes through the exhaust port as a sharp puff. This effect can 
be avoided by sufficiently reducing the steam pressure by expan- 
sion, but expansion should never be carried so far as to give a 
force on the piston (due to the steam) less than the frictional 
drag on piston and cross-head. 

The greatest items of waste in the ordinary sense of actual 
loss of heat are (a) the incomplete combustion of the fuel, and 
(b) the carrying away of great quantities of heat in the flue gases. 
The economic use of fuel for the production of mechanical power 
requires therefore a properly designed furnace and intelligent and 
careful stoking to insure complete combustion, and it requires a 
sufficient exposure of boiler surface and frequent cleaning of the 
same to facilitate the flow of heat from the hot gases into the 
boiler. 

The most pronounced sweeping process which intervenes 
between the completed combustion and the final exhaust of the 
steam from the engine is the flow of heat from the very high 
temperature of the fire in the furnace to the moderately low 
temperature of the water in the boiler, and the greatest waste 
in the operation of the steam engine in the sense of loss of availa- 
bility of heat for conversion into work is involved in this sweeping 
process and it can hardly be avoided in the steam engine because 
of the danger involved in the generation of steam at very high 
pressures (and temperatures) in a large boiler. 



374 THEORY OF HEAT. 

The best gas engines convert about thirty per cent, of the heat 
of the fuel into mechanical work. The best steam engines con- 
vert about fifteen per cent, of the fuel into mechanical work. 
The ordinary run of steam engines convert only five or eight 
per cent, of the heat of the fuel into mechanical work. 

Problems. 

53. Calculate the thermodynamic degeneration in joules per 
degree centigrade which is represented by the conversion of 
mechanical energy into heat in an electric lamp which takes 50 
watts and burns for one hour in a region where the temperature 
is o°C. Ans. 659.34 joules per degree centigrade. 

54. How much additional thermodynamic degeneration would 
be involved in the flow of the heat produced in the lamp in prob- 
lem 53 to a region at a temperature of 30 below zero centi- 
grade? Ans. 81.4 joules per degree centigrade. 

55. Assuming that thermodynamic degeneration is a true meas- 
ure of waste,* find how much waste there is in generating heat 
at a temperature of I500°C. (reckoned from ice temperature) 
in a stove and using it in a room at a temperature of 20°C, and 
find how much waste is involved in turning this heat out of 
doors at a temperature of — io° below o°C. Ans. 0.00285 joules 
per degree centigrade per joule; 0.00325 joules per degree centi- 
grade per joule. 

56. Heat is produced in a furnace under a boiler at a tempera- 
ture of I500°C. (reckoned from ice point). The boiler tempera- 
ture is i8o°C. (reckoned from ice point) and the temperature of 
the condenser of a steam engine is 70°C. (reckoned above ice 
point). How much waste is represented by the flow of 10,000 
calories of heat from furnace into boiler due to the drop in tem- 
perature and how much waste would be represented by the flow 

*Energy cannot be wasted, because, after it is converted into heat it still exists, 
or if heat flows from a region of high temperature to a region of low temperature, 
the heat is not wasted because it remains unchanged in amount. The only waste 
represented in ordinary physical processes is that which is measured by thermo- 
dynamic degeneration. 



THE SECOND LAW OF THERMODYNAMICS. 375 

of 10,000 calories from boiler to condenser due to drop in tem- 
perature? Ans. 16.44 calories per degree centigrade ; 7.07 calories 
per degree centigrade. 

57. A perfect engine taking steam at i6o°C. and exhausting 
at 70°C. would have what efficiency? A good steam engine 
working between these temperatures has efficiency equal to 75 % 
of the efficiency of a perfect engine. What is the actual efficiency 
of the steam engine? Ans. 20.8 per cent. 15.6 per cent. 

58. The mixture of air and gas in a gas engine reaches a tem- 
perature of 1 ioo°C. (reckoned above ice point) when it is exploded, 
and the temperature of this mixture is reduced to, say, 6oo°C. 
(reckoned above ice point) by expansion in the engine. What 
would be the efficiency of a perfect engine working between these 
temperatures? Ans. 36.4 per cent. 

59. An ammonia refrigerating machine is required to take 
heat from a room at — io°C. and the pipes in which the ammonia 
vapor is condensed are kept at temperature of 35°C. by being 
exposed to the air. Assuming the refrigerating machine to be a 
perfect engine, calculate the work in kilowatt-hours required to 
freeze one kilogram of ice from ice water. Assuming the re- 
frigerating machine to take 1 ^2 times as much work as a perfect 
engine, calculate the work required to freeze one kilogram of 
ice from ice water. Express the result in kilowatt-hours and in 
horse-power-hours. Ans. 0.016 kilowatt-hours; 0.024 kilo-watt 
hours; 0.0214 horse-power-hours; 0.0322 horse-power-hours. 



CHAPTER VI. 

FURTHER DEVELOPMENTS OF THERMODYNAMICS. 

[Throughout this chapter c.g.s. units are employed unless it is explicitly stated 
to the contrary; that is, pressures are expressed in dynes per square centimeter, 
volumes in cubic centimeters, masses in grams, and work and heat are both ex- 
pressed in ergs.] 

52. Specification of physical state. Watt's diagram. The physical state of a 
given fluid,* the physical state of water, for example, is completely fixed when its 
pressure and volume-per-gram are given, f It is a great help to represent the phys- 
ical state of a fluid by a point A, Fig. 28, of which the abscissa represents volume- 
per-gram and of which the ordinate represents pressure. In most of the following 
diagrams, however, abscissas represent total volumes of a given amount of the fluid and 
ordinates represent pressures. Such a diagram is called a Watt's diagram. 

When a fluid expands it does work if allowed to push against a piston; the work 
done by the fluid during an increment of volume Av is equal to p • Av, and it is rep- 
resented by the shaded area in Fig. 28. J To show that the work done by an ex- 
panding fluid is equal to p • Av, imagine the fluid to expand against a piston of 
area a. The force with which the fluid pushes on the piston is equal to pa. The 
product of this force times a small distance Ax moved by the piston is the work done 
by the expanding fluid. Therefore the work done by the expanding fluid is equal 
to pa • Ax; but a- Ax is the increment of volume of the fluid and therefore 

pa • Ax = p • Av. 

Process curve. Work done by a fluid during a process. A substance which passes 
slowly from one physical state to another because of changing pressure or changing 
volume, or both, undergoes a reversible process. The points in Watt's diagram 
which represent the successive states passed through by a fluid during a reversible 
process form a continuous§ curve called a process curve. The work done by the fluid 

*The thermodynamics of solids is very complicated. The object of this treatise 
is to bring out some of the most important facts of thermodynamics in the simplest 
possible way and therefore the details of this discussion are limited chiefly to fluids. 

fThe physical state of a fluid is also fixed when its pressure and temperature are 
given, or when its temperature and volume-per-gram are given. It is usually most 
convenient to specify physical state in terms of pressure and volume-per-gram be- 
cause of the simplicity of the expression for work done by an expanding fluid in 
terms of pressure and volume. 

}On the assumption that abscissas in Fig. 28 represent total volumes of a given 
amount of fluid. 

§Reversible processes, only, can be represented in Watt's diagram. During 
an irreversible process (a sweeping process) the fluid has no definite pressure and 

376 



FURTHER DEVELOPMENTS OF THERMODYNAMICS. 377 



during the process A B is represented by the shaded area in Fig. 29; this work 
may be done on a piston or it may be done in pushing back surrounding air, when, 
for example, water is converted into steam in an open vessel. During an increase 



axis of 
pressures 



m 



« 



_^ 



axis of 



volumes 




Fig. 28. 



Fig. 29. 



of volume work is done by the fluid, and during a decrease of volume work is done 
on the fluid. 

Cyclic process. A process is said to be cyclic when the substance comes back 
to its initial state at the end of the process. A curve in Watt's diagram which 
represents a cyclic process is a closed curve. An important example of a cyclic 
process is discussed in the next article. 

53. Carnot's cycle. A given portion of water goes through a cyclic process in 
a steam engine in which the condensed water is pumped back into the boiler. If 
each stage of this cyclic process is performed slowly without the giving of heat to 
surrounding bodies by the steam or the receiving of heat from surrounding bodies 
by the steam, and if the feed water pump is arranged to heat the water from con- 
denser temperature to boiler temperature, as explained in Art. 50, then the cyclic 
process through which the steam passes is reversible and constitutes what is called 
a Carnot cycle.* 

The water f goes through the following processes as it passes through the Carnot 
cycle: 

(a) The water is expanded at constant temperature and pressure as it evaporates 
in the boiler due to passage of some steam into the engine cylinder; 

(&) The supply of steam is then shut off from the engine and the steam in the 
cylinder is expanded, causing its temperature and pressure both to fall to the 
temperature and pressure which prevail in the condenser; 

(c) The exhaust ports are then opened and the cylinder-full of steam (at con- 
no definite temperature. If a portion of the process which carries a fluid from 
state A to state B, Fig. 29, is a sweeping process, a gap exists in the process curve 
AB. 

*So called from Sadi Carnot, the first to apply what is now called the second 
law of thermodynamics to the theory of the steam engine. 

fThis term is here intended to include both the liquid form and the vapor form. 



37% THEORY OF HEAT. 

denser pressure) is forced into the condenser by the returning piston, that is to say, 
the steam in condenser and cylinder is reduced in volume by the returning piston 
and nearly all condensed to water; and 

(d) The water from the condenser together with a certain residue of uncondensed 
exhaust steam is received by the feed-water pump and compressed. This com- 
pression causes the residue of steam to condense, thus raising the temperature of 
the feed-water up to boiler temperature as it leaves the feed-water pump. 

These four processes may be more briefly described as follows: 

(a) Expansion at constant temperature; 

(&) Further expansion without the giving of heat to the working fluid, 

(c) Compression at constant temperature, and 

(d) Further compression without the taking of heat from the working fluid. 
During process (a) a certain amount of heat H 1 is given to the working fluid 

to keep its temperature constant, and during process (c) a. certain amount of heat 
H 2 is taken from the working fluid to keep its temperature constant. If the four 
process (a), (6), (c) and (d) are performed slowly and without waste of heat to 
surrounding bodies, that is if the four processes are performed reversibly, then we 
have 

H 2 = T 2 (1) 

as explained in Art. 50. A more convenient form of this equation for subsequent 
use is 

^ = | 2 (ii) 

T x T 2 v ' 

or, considering that H x is heat given to the working fluid and that H 2 is heat taken 
from the working fluid, it is permissible to think of H l as being positive and H 2 
as being negative, so that we may write 

^+^ = da.) 

1 1 1 2 

The Carnot cycle as performed by a portion of water* in the steam engine is 
represented in Fig. 30. The line A B represents process (a), BC represents 
process (b), CD represents process (c) and DE represents process (d). The 
expansion at constant temperature which is represented by AB takes place with- 
out change of pressure because this expansion is accompanied by the evaporation 
of water in the boiler; and the compression at constant temperature which is rep- 
resented by the line CD takes place without change of pressure because of the 
condensation of the steam in the condenser. The work done by the steam on the 
piston during process (a) is represented by the area ABA'B'. The work done 
by the steam on the piston during process (&) is represented by the area BCB'C' . 
The work done by the piston on the steam, that is to say, the work given back to 
the steam during the process (c) is represented by the area CD CD', and the work 
given back to the steam during process (d) is represented by the area ADA'D'. 
Therefore the enclosed area A B CD represents the work obtained during the cycle, 
and this work is equal to H x — H 2 as explained in Art. 50. 

*This term is here intended to include both the liquid form and the vapor form. 



FURTHER DEVELOPMENTS OF THERMODYNAMICS. 379 



axis of pressures 



iL_ 



heat H t absorbed by water and 
steam 




heat H z given out by water 
steam 



A f If 




axis, of volumes 



Fig. 30. 



A gas may be made to perform a Carnot cycle. In this case process (a) (ex- 
pansion at constant temperature) takes place with falling off of pressure because 
the pressure of a gas is inversely proportional to the volume when the temperature 
is constant; and process (b) (compression at constant temperature) takes place 
with rise of pressure for the same reason. Fig. 31 represents a Carnot cycle as 
performed by a gas. 

A curve showing the relationship between volume and pressure of a gas at con- 
stant temperature is called an isothermal curve; the curves a and c in Fig. 31 are 




axis of volumes 



Fig. 31. 



3 8o 



THEORY OF HEAT. 



isothermal curves corresponding to temperatures T x and T 2 . A curve which rep- 
resents the relation between volume and pressure of a gas when it expands without 
receiving or giving off heat is called an adiabatic curve (or isentropic curve) ; curves 
b and d in Fig. 31 are portions of the adiabatic curves <f> x and 4> 2 . 

54. Clausius's theorem. Entropy of a substance. Consider a fluid which, 
starting from any given pressure p and volume v as shown in Fig. 32, is subjected 



axis of pressures 




Fig. 32. 



to any combination whatever of slow heating (or cooling) and expanding (or con- 
tracting) so that its changing pressure and volume may be represented by the 
successive ordinates of the closed curve CC. The process undergone by the fluid 
is reversible because it is performed very slowly, and it is a cyclic process because 
the substance comes back to its initial state. Consider two adiabatic curves <p x 
and 2 ver y c l° se together, and consider the cyclic process which is represented by 
the heavy lines a, b, c, and d. This process constitutes sensibly a Carnot cycle 
because the portions a and c of the curve are so short that any slight change of 
temperature which takes place while the fluid is passing over these portions is 
negligible. Let AH 1 be the heat given to the fluid while it is passing over the 
short curve a, and let T l be the temperature of the fluid during this process; let 
AH 2 be the heat taken from the fluid while the fluid is passing over the short curve 
c, and let T 2 be the temperature of the fluid during this process. Then, according 

to equation (180), we have 

AHj_ . AH 2 



T x + T 2 



CD 



in which AH V being heat delivered to the fluid, is considered as positive, and AH 2 , 
being heat taken from the fluid, is considered negative. 

The whole of the given cyclic process may be broken up into pairs of corre- 



FURTHER DEVELOPMENTS OF THERMODYNAMICS. 3 8 * 

sponding parts like a and c, Fig. 32, so that the entire heat taken in by the fluid and 
given out by the fluid during the different parts of the given cyclic process may be 
arranged in pairs like AH X and AH 2 , and each pair of heat-parts satisfies an 
equation like equation (i). Therefore the sum of all quotients obtained by dividing 
the heat delivered to the fluid (heat taken from the fluid being considered as negative) 
at each step of any reversible cyclic process by the absolute temperature of the fluid at 
the step is equal to zero. That is 

2^ = o (186) 

Consider two states of thermal equilibrium of a fluid A and B, Fig. 33, p and v 
being the pressure and volume of the fluid in state A, and p' and v' being the pressure 
and volume of the fluid in state B. Let the curves aa and bb Fig. 33 represent 





axis of 


pressures 
v f 


B 




V 




rP 


vf 




A 


f - — — . ^ 
b 

P 










axis of volumes 





Fig. 33- 



any two different reversible processes leading from A to B. Then process bb re- 
versed and process aa together constitute a cyclic process starting from A and 
returning to A. Therefore the sum 2AH/r is equal to zero when it is extended 
over process aa and over process bb reversed. Therefore the sum 2AH/T" ex- 
tended over process aa is equal and opposite to the sum 2A H»/ T extended over 
process bb reversed, but the sum 2AH/T extended over process bb reversed is 
equal and opposite to SAff/r extended over process bb not reversed, because 
heat given to a fluid (or taken from a fluid) during any step of a given process be- 
comes heat taken from the fluid (or heat given to the fluid) if the process is reversed; 
that is to say, AH for each step of a reversed process is opposite in sign to the 
corresponding value of AH for the process not reversed. 

The above argument leads at once to the proposition that 2AH/T has the 
same value for any two, and therefore for all reversible processes which lead from one 
given state of thermal equilibrium A to another given state of thermal equilibrium B. 
This theorem is due to Clausius. 

Consider a portion of heat A H given to (or taken from) a substance during a 
small part of a reversible process and imagine this heat to be produced by the degen- 



3^2 THEORY OF HEAT. 

eration of work into heat (or if A H is heat taken from the substance imagine it 
to be regenerated into work). Then AH/T would be the thermodynamic degen- 
eration (or regeneration) associated with the small part of the reversible process, 
due, of course, not to turbulence in the substance which is undergoing the reversible 
process but to the actions outside of the substance which are involved in the assumed 
production of AH by degeneration of work into heat (or the actions which are 
involved in the regeneration of AH into work). Therefore the sum 2A H/T" for 
any reversible process like aa or bb Fig. 33 which carries a fluid from a given 
state A to a given state B is a measure of the thermodynamic degeneration which is 
associated with the difference in state between A and B. This thermodynamic de- 
generation which is associated with a difference in state is called the entropy differ- 
ence between the two states. Thus, if state B can be reached from A by a sweeping 
process, then 2 A H/T has a positive value for any reversible process leading from 
A to B and this value of 2AH/ T is called the entropy of the state B referred to 
the state A. 

Let <f> be the entropy difference between two states of a substance, then according 
to the above definition of entropy difference we have 

<£ = 2^? (I9) 

where AHjT refers to an element of a reversible process leading from the one 
state of the substance to the other, AH. being the heat delivered to the substance 
during this element or step and T being the temperature of the substance during 
the step. If the two states are very near together then equation (19) may be 
written in the form 

Acj> = AH/T (20) 

in which A0 is the entropy difference between the two adjacent states of a sub- 
stance, AH is the heat which must be delivered to the substance to change it from 
one state to the other, and T is the absolute temperature of the substance.* 

Entropy differences, only, have a physical significance. A certain state of fluid 
may be arbitrarily chosen as a zero state or reference state, and the value of the 
entropy of the substance in any other given state may be defined as the value of 
SAH/T" extended over any reversible process leading from the zero or reference 
state to the given state. This is equivalent to assigning arbitrarily the value zero 
to the entropy of the substance in the zero state. 

Example. A gas is slowly expanded under a piston at constant temperature T, 
and a quantity of heat H is given to the gas to keep its temperature constant- 
Then, since T is constant during the process, H/ T is the increase of entropy of 
the gas during the expansion. That is to say there is an entropy difference of 
H/T between the initial and final states of the gas. The same change of state 
may be brought about by allowing the gas to expand freely (through an orifice), 
and the thermodynamic degeneration involved in this sweeping process is measured 
by the entropy difference between the initial and final states of the gas. 

The integral 2 • A Hj T cannot be applied to a sweeping process because a sub- 

*Of course the temperature in one state may differ from the temperature in the 
other state by an infinitesimal amount. 



FURTHER DEVELOPMENTS OF THERMODYNAMICS. 383 

stance which is undergoing a sweeping process is not at any time in thermal equi- 
librium during the process and the substance therefore has no temperature during 
the process. The thermodynamic degeneration which is involved in a simple sweep 
is the entropy difference between the initial and final states of the substance, and 
it can be evaluated only by calculating the value of the integral S-Afl/r during 
a reversible process which leads from the same initial to the same final state of the 
substance. 

The entropy of a substance in a given state is proportional to the mass of the sub- 
stance. This is evident when we consider that the value of AH for every step 
of a reversible process is doubled if the mass of the fluid is doubled. Entropy is 
expressed in units of heat per degree of temperature as explained in Art. 48. 

55. Specific heats of a gas.*' The number of thermal units (ergs) required to 
raise the temperature of one gram of a gas one degree is called the specific heat- of 
the gas. If the volume of the gas does not change during the rise of temperature, 
then, inasmuch as no external work is done by the gas, all of the heat applied goes 
to increase the internal energy of the gas. If, however, the gas be allowed to expand 
as the temperature rises, to such an extent, for example, as to keep the pressure 
constant, then the heat which is supplied to the gas to raise its temperature must 
not only increase the internal energy of the gas but must also make up for the ex- 
ternal work done by the gas as it expands. The specific heat of a gas has therefore 
two important values, namely, the speicfic heat at constant volume C v , and the 
specific heat at constant pressure C P , of which the latter has the larger value. 

Relation between Cv and C P . To raise the temperature of M grams of a gas A T 
degrees requires CvM • A T units of heat (ergs) if the volume is kept constant. If 
the gas is then allowed to expand by an amount Ay sufficient to bring the pressure 
back to its initial value p, an amount of work equal to p - Av is done by the ex- 
panding gas and an amount of heat equal to p • Av must be given to the gas to 
keep up its temperature, for, according to Joule and Thomson's principle, the only 
appreciable cause of change of temperature by expansion of a gas is the loss of 
energy of the gas by the doing of external work. Therefore a total amount of heat 
equal to CvM • AT-\-p • Av is required to raise the temperature of the gas by the 
amount AT" when the gas is allowed to expand sufficiently to keep its pressure 
constant; but according to the definition of C P the amount of heat required for 
this change of temperature at constant pressure is equal to C P M • A T. Therefore 

CpM -AT=C v M-AT+p-Av (i) 

*Not only does a gas have two specific heats according as it is kept at constant 
volume or at constant pressure during the increase of temperature, but a gas has two 
important values for its bulk modulus (see Mechanics, Art. 99). If the temperature 
of the gas is kept constant during compression by the extraction of heat from the 
gas, then the rise of pressure due to a given decrease of volume is less than if no 
heat is abstracted from the gas during compression. In the one case we have what 
is called the isothermic bulk modulus, and in the other case we have what is called 
the isentropic or adiabatic bulk modulus. 

All substances have two values of specific heat, namely, a specific heat at con- 
stant volume and a specific heat at constant pressure, and every substance has 
isothermic and isentropic moduli of elasticity. It is only in the case of gases, how- 
ever, that these differences are large enough to be appreciable. 



3^4 



THEORY OF HEAT. 



Proposition. The constant R in equation (4b) of Art. 6 for a given gas is equal 
to the difference between the specific heat of the gas at constant pressure (C P ) 
and the specific heat of the gas at constant volume (C«); that is 

C P -Cv=R (21) 

Proof. Imagine M grams of the gas to have been increased in temperature by 
the amount Ar at constant pressure, Av being the increment of volume. Then 
we have equation (i) above. The increment of volume necessary to keep the 
pressure constant when the temperature increases is found by differentiating equa- 
tion (4&) remembering that T and v, only, vary. This gives 



p- Av = MR AT 



(ii) 



Substituting this value of p • Av in equation (i), we have equation (21). 
56. Isothermic expansion and compression. When the temperature of a gas 
is kept at a constant value by supplying heat to the gas as the gas expands, or by 



axis of pressures 




axis of 



Fig. 34- 



volumes 



abstracting heat from the gas as the gas is compressed, the expansion or compres- 
sion is said to be isothermic. Equation (46) gives the relation between p and v 
of a perfect gas during isothermic expansion or compression, T being then a con- 
stant as well as M and R. 

The full-line curves in Fig. 34 show graphically the isothermic relations between 
p and v for air for several different temperatures. The ordinates of these curves 
represent pressures and the abscissas represent volumes. The curves are sometimes 
called the isothermal lines of the gas; they are equilateral hyperbolas. 

57. Adiabatic or isentropic expansion and compression. When a gas is ex- 
panded or compressed and not allowed to receive heat from, or to give heat to sur- 



FURTHER DEVELOPMENTS OF THERMODYNAMICS. 385 

rounding bodies, the expansion or compression is called adiabatic or isentropic.*- 
The relation between p and v of a perfect gas during isentropic expansion or 
compression is: 

pv k = a. constant (22) 

in which 

k= % (23) 

The dotted curves in Fig. 34 show the isentropic relation between p and v for 
air (for two different values of the entropy 4>i and 2 ). The value of k for air is 1.41. 

Proof of equation (22). Consider a very slight isentropic expansion of a gas, and 
let Av, Ap and Ar be the actual changes of volume, pressure and temperature; 
Av being an increment, while Ap and A T are both decrements and therefore both 
negative. 

The work p ■ Av done by the gas is all made up by the decrement MC V • AT 
of the internal energy of the gas;f therefore, remembering that MCv • Ar is nega- 
tive while p ■ Ay is positive we have : 

p-Av=-MCv'AT (i) 

From the equation pv = MRT (46) we have: 

t ~Wr*° (ii) 

whence 

A T = ~ R (p -Av+v Ap) (iii) 

Substituting this value of AT in equation (i), remembering that R= C P —C V 
according to equation (21), and that k=C P /Cv (23), we have: 

Ap Av ... 

— + k — = o (iv) 

p V 

whence by integrating 

Log p-\-k Log v = sl constant 
or 

Log (pv k ) = a constant 
or 

pv k = 3. constant 

58. Proposition. Temperature ratios as measured by a gas thermometer conform 
to temperature ratios as defined by equation (15) of Art. 50 if the gas which is used in 
the gas thermometer conforms to Boyle's Law, and if it is a gas of which the temperature 
neither rises nor falls during free expansion. To establish this proposition, it is 
necessary to consider very carefully what assumptions and what experimental facts 
lie at the basis of the discussion in Arts. 55, 56 and 57. 

(1) Some method of measuring temperature is necessary as a basis for the ex- 

*The entropy of a gas is constant during this kind of expansion. 

fThis is equivalent to saying that the only cause of cooling of the expanded gas 
is its loss of energy by the doing of external work p • Av. See discussion of equation 
(21) in Art. 55. 
26 



3 86 



THEORY OF HEAT. 



perimental determination of specific heats. In fact all accurate work in specific 
heats has been based upon temperatures as measured by the gas thermometer. 
The use of equation (4b) in Arts. 55, 56, and 57 means, in the first place, that the 
gas under discussion is assumed to conform to Boyle's law and it means, in the 
second place, that temperatures as used in the discussion are supposed to be 
measured by a gas thermometer. 

(2) The assumption is made in Art. 55 that the gas under discussion is one of 
which the temperature neither rises nor falls during free expansion and this as- 
sumption again appears in Art. 57. 

(3) Regnault's experiments on the specific heats of gases show that the specific 
heat at constant pressure of any ordinary gas does not change sensibly with the 
temperature or pressure of the gas within a moderate range of temperatures and 
pressures. Therefore C P may be treated as a constant. Therefore the specific 

axis of pressures 




axis of volumes 



Fig. 35- 



heat of any ordinary gas at constant volume, C», is sensibly constant for ordinary 
temperatures and pressures according to equation (21). In fact, the integration 
of equation (iv) in Art. 57 assumes the constancy of k (=C P /Cv). 

To establish the above proposition consider a Carnot cycle a b c and d, between 
two very closely adjacent adiabatic curves 4>\ and 2 and between any two iso- 
thermal curves T x and T 2 as shown in Fig. 35- The proposition is then established 

if we can show that 

AH A= T 1 (i) 

AH 2 T 2 

where A H x is the heat taken in by the gas during process a, and A H 2 is the heat 

given out by the gas during process c in Fig. 35. 



FURTHER DEVELOPMENTS OF THERMODYNAMICS. 387 

The gas being one of which the temperature neither rises nor falls during free 
expansion, AH X is equal to the work done by the gas during process a, and AH 2 
is equal to the work done on the gas during process c, and inasmuch as the two 
adiabatic curves <p x and 2 are very close together, this work is in each case equal 
to the change of volume (A^ or Av 2 ) multiplied by the prevailing pressure (£] 
or p 2 ). It remains therefore to find an expression for the increment of volume of 
a gas along any isothermal curve T between the two adiabatic curves <P\ and 2 * 

The equation of one of the isothermal lines in Fig. 35 is 

pv = MRT (ii) 

according to equation (46), and the equation to the adiabatic lines is 

pv k =C (iii) 

according to equation (22); and to go from one adiabatic curve to another is to 
assign a definite increment AC to the quantity C. Differentiating equation (iii) 
we have 

A C = kpv*~ l • A v + V h • Ap (iv) 

and in order that the two variations Av and Ap may correspond to a movement 
along an isothermal line, they must be variations obtained from the differentiation 
of equation (ii) on the assumption that T is constant. Therefore, differentiating 
equation (ii), we have 

MRT . 

Ap = r Ay 

v 

or, using pv for MRT, we have 

i> 
Ap = — - • A v (v) 

v 

Substituting this value of Ap in equation (iv) and solving for Av, we have 

A V = ^— (vi) 

{k-i)pv k ~ l 

Therefore the work p • Av done on the gas while it is expanding along an isothermal 
curve from one adiabatic line to the other adiabatic line in Fig. 35 is 

±h = (h A( ;,. 1 (vii) 

ik — i)v k 1 

in which Aff is written for the work done by the gas because the work done by 
the gas is equal to the heat which is imparted to the gas to keep its temperature 
constant. This expression for AH may be written 

AH = pv — r 

which may be written 

AC 

AH = MRT'~ —5 (viii) 

{k — i)pv k 

inasmuch as pv is equal to MRT. With respect to this equation (viii) it is to 
be noted that the numerator has a definite value for a movement from one to another 
adiabatic curve, and that the denominator has the same value everywhere along 



3^8 THEORY OF HEAT. 

any adiabatic curve. Therefore the fraction Ac/[(k—i) pv k ] has the same value 
during processes a and b in Fig. 35. Writing T x for T in equation (viii) we have 
an expression for AH X and writing T 2 for T in equation (viii) we have an expression 
for AH 2 , and by combining these two results we find at once that Affj/Aff 2 
equals T x / T 2 . 

59. Rise in temperature of a gas during isentropic compression. We may 

eliminate p (or v) from equation (22) by substituting the value of p (or v) from 
equation (4b). In this way we find: 

Tv k ~ l = a constant (24) 

or 

T pd-k)/k _ a cons tant (25) 

during isentropic expansion or compression. 

Examples of isentropic expansion and compression. The expansion and com- 
pression of the air in sound waves is isentropic, because the expansion and com- 
pression take place so quickly that the expanded or compressed portion of the air 
does not have time to give off or to receive any appreciable amount of heat. 

When a large mass of air moves upwards, as, for example, when a wind blows 
up a mountain slope, the pressure of the mass of air falls off. with increasing altitude 
and the temperature is reduced. The expansion is isentropic, inasmuch as the 
mass of air is so large that it cannot, during the brief time of the ascent, receive 
or give off an appreciable amount of heat. Suppose that the rising mass of air 
was initially at temperature T x and pressure p lt and that its pressure has fallen 
to p 2 . The temperature T 2 corresponding to p 2 may be found from equation (25), 
which gives 

T x pf-W = T 2 p<P~ k V k (26) 

A very interesting phenomenon due to the cooling of a rising column of air by 
isentropic expansion is the formation of the beautiful cumulous clouds on a quiet 
summer day. The warm moist air near the ground starts streaming upwards 
through the superimposed cold air. This upward stream once started draws like 
a chimney and the rising column develops until it becomes very large and very 
high. At a very sharply defined altitude the pressure reaches a value for which 
the temperature [according to equation (26)] of the rising air is reduced to the dew 
point. The further cooling which is produced as the air passes above this level, 
causes the condensation of water vapor and the formation of mist or cloud. The 
strikingly fiat bottom of a cumulous cloud is at that altitude where the rising 
air reaches the temperature of the dew point. 

The condensation of water vapor as rain is due very largely to the isentropic 
cooling of the great rising column of air, sometimes hundreds of miles in diameter, 
at the center of what is technically called a cyclone (See Art. 10). 

When a gas is quickly compressed under a piston in a cylinder, the compression 
is isentropic. If the initial volume is v x , initial temperature T x , and final volume v 2 , 
we may find the temperature T 2 corresponding to v 2 from equation (24), which gives 

T x v x k ~ l = T 2 v 2 k ~ l (27) 

60. Method of Clement and Desormes for the experimental determination of k. 

A vessel filled with the gas at pressure p x and temperature T x is opened to the air. 



FURTHER DEVELOPMENTS OF THERMODYNAMICS. 389 

allowing the pressure to drop quickly to atmospheric pressure p 2 , the temperature 
falling at the same time to T 2 . Equation (26) then gives 

TiPi{ i-Wk = Ti p 2 {i-k)/k (26) bis 

The vessel is then quickly closed and allowed to stand until it has reached its 
former temperature T x , when the pressure is p$, so that from Gay Lussac's law 
we have 

P2 Pt 

Substituting the value of T 2 from (i) in equation (26) and reducing, we have 

?=*? (ii) 



(£)'- 



Pi 

from which k may be calculated, p v p 2 , and p z having been observed. The value 
of k for air is 1.41. 

Remark. The numerical value of the constant R equation (4&), is easily cal- 
culated from the observed value of the density Mjv of the gas at a known tem- 
perature T and pressure p. Therefore the value of (C P —Cv) is known for the 
gas from equation (21). The difference (C P — Cv) being thus known and the ratio 
C P /Cv being determined by Clement and Desormes' method, one may easily cal- 
culate the values C P and Cv. 

61. Entropy of one gram of gas expressed in terms of its pressure and volume. 
An expression for the entropy of one gram of a gas in terms of its pressure and vol- 
ume may be derived with the help of the relation 

' **-¥ a) 

in which A0 is the increase of entropy of the substance when an amount of heat 
AH is given to the substance at temperature T. The meaning of this equation 
may be understood by a careful study of Art. 54. 

The amount of gas under consideration being one gram, equation (46) becomes 

pv = RT (ii) 

so that the change of temperature AT due to a given change of volume Av at 
constant pressure is found by differentiating equation (ii) with respect to v, giving 

AT = ^ (in) 

R 

The change of temperature AT corresponding to a change of pressure Ap at 
constant volume is found by differentiating equation (ii) with respect to p, giving: 

AT = V -^ (iv) 

R 

Change of entropy due to change of volume at constant pressure. Equation (iii) 
gives the change of temperature produced by change of volume Av at constant 
pressure, and an amount of heat, AH— Cp AT, must be given to the gas to keep 



390 THEORY OF HEAT. 

the pressure constant during this change of volume. Therefore using the value of 
AT from equation (hi) we have 

AH= C P -pAv 
R 

and substituting this value of AH in equation (i) we have 

A0 = C P ■ — (v) 

v 
in which pv has been used for RT. 

Change of entropy due to change of pressure at constant volume. Equation (iv) 

gives the change of temperature due to change of pressure Ap at constant volume 

and an amount of heat, AH=C -AT, must be given to the gas to produce the 

increase of pressure at constant volume. Therefore using the value of Ar from 

equation (iv) we have *. 

and substituting this value of AH in equation (i) we have 

A0 = C v • ^ (vi) 

P 
in which pv has been used for RT. 

Complete differential of entropy. Equation (v) expresses the differential of en- 
tropy with respect to volume, and equation (vi) expresses the differential of entropy 
with respect to pressure; therefore the complete differential of entropy is given 
by the equation 

A<P=C P — + CV^ (vii) 

v p 

which by integration gives 

<p = C P logc v + Cv loge p (28) 

from which the constant of integration is omitted inasmuch as differences of entropy, 
only, have physical significance as explained in Arts. 53 and 54. 

62. Application of equation (15) to the vaporization of water. Consider one 
gram of water in the form of saturated steam at temperature T, the pressure and 
volume of the steam being p and V s , respectively, as represented by the co-ordi- 
nates of the point a in Fig. 36. Imagine this one gram of water to be carried through 
a Carnot cycle as follows: 

(1) Compress the steam at constant temperature T until it is nearly all con- 
densed into water at the point b in Fig. 36. During this condensation an amount 
of heat H 2 is abstracted from the steam. 

(2) Compress the condensed water at point b (together with an infinitesimal 
residue of uncondensed steam), without abstracting heat from it, until its temper- 
ature and pressure rise to r+Ar and P+Ap, respectively, as represented by the 
co-ordinates of the point c in Fig. 36. 

(3) Allow the water to be converted into steam by increasing its volume, the 
temperature being kept at T-\-AT by giving an amount of heat H x to the water. 
This process of vaporization is represented by the line cd in Fig. 36. 

(4) Allow the steam (together with an infinitesimal residue of unvaporized 



FURTHER DEVELOPMENTS OF THERMODYNAMICS. 391 

water) to expand, without receiving heat, until its temperature and pressure fall to 
the initial temperature and pressure. This process is represented by the line da 
in Fig. 36. 

The heat H 2 taken from the water in part 1 of the cycle is equal to the latent 
heat of vaporization L of water at the given temperature and pressure because it 
is the amount of heat given off by one gram of steam in being condensed to water 
at a fixed temperature; and the heat H x given to the water during part 3 of the 



axis of pressures 




volumes 



Fig. 36- 

cycle exceeds Hi by an amount which is equal to the work AW gained during the 
cycle; this work is represented by the shaded area in Fig. 36 and it is equal to 
(V s —Vw) • Ap, where Vw is the volume of one gram of liquid water at temper- 
ature T. Therefore, using L instead of H2, equation (15) gives 



L+(Vs 



whence 



AT 



Vw) - Ap 
L 



T+ AT 



T( V s - V w ) 



(29) 



This equation expresses the steepness (Ap/AT) of the vapor-pressure curve 
in terms of the latent heat of vaporization L of the liquid, the absolute tem- 
perature T of the boiling point, and the increase of volume of one gram of the 
liquid when it is converted into vapor; the vapor-pressure curve being a curve 
of which abscissas represent boiling points (temperatures) and ordinates represent 
corresponding vapor pressures. 

Equation (29) applies not only to vaporization but also to freezing. It is worth 
while, however, to discuss the case of freezing independently as follows. Consider 



392 



THEORY OF HEAT. 



one gram of water initially in the form of ice at temperature T, the pressure and 
volume of the one gram of ice being p and Vi, respectively, as represented by the 
co-ordinates of the point a in Fig. 37. Imagine this one gram of water to be carried 
through a Carnot cycle as follows: 

(1) Impart an amount of heat H x to the water (ice) until nearly all of the 
ice is melted at temperature T. This melting is accompanied by a great decrease 
of volume, and the process is represented by the line ab in Fig. 37. 

(2) Increase the pressure by the amount Ap, without giving heat to or taking 
heat from the substance. This increase of pressure causes a decrease of the freezing 
point from T to T — AT* the infinitesimal residue of ice melts, and the heat absorbed 





curia of 
















pressures 


% 


- . 










c 


freezing at T—AT 


d 






mMw////mw/A 


Wfc. 


a 


-IK- 

',AP 
— *- 






* melting at T 




a 








H < 
















v i 








\P 




\r 


--* 


axis~iiT\_ 




















volumes J 








I 

















Fig. 37- 

in this melting lowers the temperature of the whole to T—AT. This process is 
represented by the curve be in Fig. 37. 

(3) Abstract an amount of heat H 2 from the substance until nearly all of the 
water is frozen at temperature T — AT. This freezing is accompanied by a great 
increase of volume, and the process is represented by the curve ed in Fig. 37. 

(4) Decrease the pressure to the original value without giving heat to or taking 
heat from the substance. This change of pressure causes the infinitesimal residue 
of water to be frozen, and the heat thus liberated raises the temperature of the sub- 
stance from T—AT to T. This process is represented by the curve da in Fig. 37. 
The amount of heat H x is equal to the latent heat of fusion L of ice, and the amount 
of heat ff 2 is equal to H l —A W, where AjFisthe work gained during the cycle; 



*This statement that Ar is a decrease of temperature must be kept in mind 
when one comes to interpret the final equation (30). It is more convenient to 
think of a decrease of temperature as being negative, and therefore the negative 
sign should be inserted in equation (30) giving equation (31). 



FURTHER DEVELOPMENTS OF THERMODYNAMICS. 393 



and this work is represented by the shaded area in Fig. 37 

tion (15), we have 

L _ Hi __ T 

L-(Vi- V w )-Ap ~ H 2 " 

or 

AT T 

75- z"*-™ 



Therefore from equa- 



T — AT 



(30) 



in which AT is the decrease of freezing point due to an increase of pressure Ap, 
But A T is a decrease of temperature, and it is to be considered as negative so that 
equation (30) may be more properly written in the form 

AT 



T 



Vw) 



(31) 



It is interesting to use this equation for calculating the decrease of freezing 
point of water due to an increase of pressure from one atmosphere to two atmos- 
pheres. Under these conditions, Ap = one atmosphere (equals, in round numbers, 
1,000,000 dynes per square centimeter). T is equal to 273 degrees, L is equal to 
80 calories (or 80 X4.2X10" ergs), Vw equals 1. 000127 cubic centimeters, and F, 
equals 1. 090821 cubic centimeters; therefore equation (31) gives Ar equal to 
0.01338 centigrade degree. 

63. Discussion of thermo-elastic properties of rubber. Take the ends of a 
rubber band in the hands, hold the band against the lips (which are very sensitive 
to changes of temperature), stretch the band and notice that its temperature rises, 



axis of tensions 




axis of 



elongations 



correct diagram of rubber engine 

Fig. 38. 

shorten the band and notice that its temperature falls. From these observed facts, 
it is immediately evident that heat has to be abstracted from a rubber band to keep 



394 



THEORY OF HEAT. 



this temperature constant while it is being stretched, and that heat has to be given to a 
rubber band to keep its temperature constant while it is being shortened. 

Starting from these facts, it can be shown by argument that a rubber band which 
is stretched by hanging a given weight upon it is shortened when its temperature is 
raised slightly, or, what amounts to the same thing, // the band is prevented from 
shortening its tension increases when its temperature is raised slightly. These results 
will be assumed as the basis of our argument and the argument will show that 
they are necessarily true. 

When a rubber band is elongated, the tension of the band increases and a curve 
may be plotted of which the abscissas represent elongations and of which the or- 
dinates represent tensions. Thus, the abscissas of the curve ab in Fig. 38 repre- 
sent elongations and the corresponding ordinates represent tensions of the rubber 
band at constant temperature TV 

Starting at any point a in Fig. 38 with the rubber band -at temperature T2, let 
the band be carried through a four-stage cycle as follows: 




impossible diagram 9>t rubber engine 

Fig. 39- 



(1) Increase the length of the band at constant temperature Ti as represented 
by the line ab. During this process a definite amount of heat Hi must be abstracted 
from the band. 

(2) Continue to stretch the band, without abstracting heat from it, until its 
temperature rises to T x . This part of the cycle is represented by the line be in 
Fig. 38. 

(3) Allow the band to shorten at constant temperature T x as represented by 
the line cd. During this process a definite amount of heat H x must be given to 
the band to keep its temperature constant. 



FURTHER DEVELOPMENTS OF THERMODYNAMICS. 395 

(4) Allow the band to continue to shorten, without giving heat to it, until its 
temperature falls to T2. 

The net result of this cycle of operations is that a certain amount of heat H x 
is taken by the band from a region at high temperature T x and a certain amount of 
heat Hi is delivered by the band to a region at low temperature Ti. But the op- 
erations (1), (2), (3) and (4) as above described are reversible, and reversible op- 
erations cannot bring about thermodynamic degeneration, therefore the transfer of 
a portion of the heat H x to a lower temperature must be compensated by a con- 
version of the remainder of this heat into work, that is to say, the work done by 
the band as it shortens at temperature T x must be greater than the work done on 
the band in stretching it at temperature Ti, or in other words, the tension eg must 
be greater than the tension ef as shown in Fig. 38. Given a rubber band under 
tension ef at temperature T2, if the elongation remains unchanged, the tension 
must rise to eg when the temperature is raised from T x to T2, or, what amounts 



warm 



warm 




cold' 



Fig. 40. 



to the same thing, if the tension of the rubber band is kept constant, as, for example, 
by means of a weight hung upon it, then the increase of temperature from Ti to 
T x must produce a shortening of the band from Oe to Oe'. If the tension of 
a rubber band were greater at the lower temperature T 2 than at the higher tem- 
perature 7\ as shown in Fig. 39, then to carry the band around the cycle abed would 
result in (a) the doing of work on the band (the amount of work so done 
being converted into heat) and (b) the transfer of heat from the high tem- 
perature region T x to the low temperature region T 2 both of which effects 
would constitute thermodynamic degeneration. The process abed is, however, 



39 6 THEORY OF HEAT. 

irreversible and cannot lead to thermodynamic degeneration. Therefore eg must 
be greater than ef. 

Figure 40 shows the essential features of a rubber engine. The two wheels WW 
and ww lie in one plane and the axles of the wheels are geared together so that they 
must both rotate at the same speed in the same direction, bands of rubber rrr are 
stretched between the rims of the two wheels, one side of the whole apparatus is 
covered by a hood and kept warm, and the other side is exposed to the cool air. 
The greater tension of the warm rubber bands causes rotation in the direction of 
the curved arrows.* 

Problems. 

60. Pressure in Watt's diagram is represented by the ordinates to the scale 10 
pounds per square inch of pressure per inch of ordinate, and volumes are represented 
by abscissas to the scale 100 cubic inches of volume per inch of abscissa. What 
amount of work in foot-pounds is represented by each square inch of area under a 
process curve? Ans. 83.3 foot-pounds. 

61. Calculate the entropy in calories per degree centigrade of 10 grams of water 
in the form of saturated steam at ioo°C. Base the calculation upon assumed 
constancy of specific heat of water from zero to ioo° taking ice water as the zero 
state. Ans. 17.6 calories per degree centigrade. 

Note. In solving this problem use algebraic integration or divide the rise of 
temperature from o°C. to ioo°C. into, say, 20 equal steps and calculate the quotient 
AH/T for each step. 

62. A cylinder 12 inches long (in the clear) and 10 inches in diameter contains 
air at a pressure of 15 pounds per square inch. The air is compressed isothermally. 
Plot on cross section paper the curve showing the pressure for each position of the 
moving piston while the piston moves from a point 12 inches from the end of the 
cylinder to a point 6 inches from the end of the cylinder and calculate the work in 
foot-pounds done on the air during the compression. Ans. 815 foot-pounds. 

Note. Calculate the work by determining the area under the plotted curve. 
To determine this area use the process of algebraic integration or divide the area 
up into a number of vertical strips and calculate the approximate area of each by 
considering the intervening portion of the curve as a straight line. 

63. The gas in the above cylinder is compressed adiabatically. Plot the curve 
showing the pressure for each position of the piston in moving from the point 12 
inches from the end of the cylinder to a point 6 inches from the end of the cylinder 
and calculate the work in foot-pounds done during the compression. Ans. 944 
foot-pounds. 

64. A perfect engine takes compressed air at 25°C. and 150 pounds per square 
inch pressure (above atmospheric pressure). At what cut-off must the engine be 
adjusted in order that the temperature of the expanded air may be — 2o°C? Ans. 
67.1 per cent. 

*A series of interesting applications of the second law of thermodynamics is 
given in an important paper entitled "The Function of Osmotic Pressure in the 
Analogy between Solutions and Gases," by J. Van't Hoff, Philosophical Magazine, 
Vol. 26, pages 81-105, August, 1888. 



FURTHER DEVELOPMENTS OF THERMODYNAMICS. 397 

Note. Assume the expansion in the cylinder to be adiabatic. 

65. A bicycle pump is full of air at 15 pounds per square inch, length of stroke 
15 inches. At what part of the stroke does air begin to enter the tire at 40 pounds 
per square inch above atmospheric pressure on the assumption that the compression 
takes place without rise of temperature? Ans. 4. 113 inches from end of stroke. 

66. Air at 15 pounds per square inch pressure and at 20°C. is pumped into a 
bicycle tire at a pressure of 40 pounds per square inch (above atmospheric pressure). 
Find the temperature of the compressed air, assuming the compression to be adia- 
batic, and find at what part of the stroke the air reaches 40 pounds per square inch 
above atmospheric pressure. Ans. i53°.5C; 0.6 of stroke. 

67. Air at 200 pounds per square inch (above atmospheric pressure) is used to 
drive an air engine which exhausts at 15 pounds per square inch (above atmospheric 
pressure). Required the temperature of the high pressure air in order that there 
may be no possibility of frost forming in the exhaust ports of the engine, the ex- 
pansion of the air in the engine being assumed to be adiabatic. Ans. 2i2°C. 

Note. Frost frequently forms in the exhaust ports of an air-driven engine. 
This occurs when the air (moist) is cooled below o°C. by the expansion which 
takes place in the engine. 

68. The atmospheric pressure at the ground is 755 millimeters. At a distance 
of 2,000 feet above the ground the pressure is 695 millimeters. The temperature 
of the air at the ground is 3i°C. Find the value of the dew point of the air at the 
ground in order that a rising column of air would form a cloud at 2,000 feet above 
the ground. Ans. 25°.5 C. 

Note. In this problem neglect the influence of the water vapor upon the law 
of adiabatic expansion of the air. Note that the pressure of the water vapor is 
decreased in the ratio 755 to 695 so that the air will have to be cooled below the 
temperature which expresses the dew point at the ground. 

The following table showing relation between dew point and vapor pressure is 
needed in the solution of this problem. 

Dew Point. Vapor Pressure. 
23°C. 20.9 mm. 

24°C. 22.2 mm. 

25°C. 23.5 mm. 

26°C. 25.0 mm. 



CHAPTER VII. 

TRANSFER OF HEAT. 

64. Conduction; convection; radiation. There are three quite 
distinct processes* by means of which heat is transferred from 
one place to another. 

Conduction. If one end of a metal rod is held in a flame the 
whole rod becomes eventually more or less heated. Heat is 
transferred along the rod by being handed on from one part of 
the rod to the next part beyond. This mode of transfer of heat 
is called heat conduction. When one end of a copper rod is 
placed in a flame, the whole rod becomes heated in a very short 
time, heat spreads along an iron rod more slowly than along a 
copper rod, and the spread of heat along a glass rod, one end of 
which is held in a flame, is scarcely perceptible. Thus, copper is 
said to be a good heat conductor, iron not so good, and glass is a 
poor heat conductor. A very poor heat conductor is sometimes 
called a heat insulator. Porous substances such as saw-dust and 
wool are good heat insulators. 

Convection. Heat may be transferred from one place to an- 
other by the flow of a hot fluid. Thus in a heating plant, heat 
flows from the furnace into the boiler by conduction and is then 
carried to the various parts of the building by hot water or steam. 
Great quantities of heat are carried by the winds and ocean cur- 
rents from one place on the earth to another. This mode of 
heat transfer is called heat convection (see Art. 10). 

Radiation. Heat is transferred from a hot to a cold substance 
through intervening space even when this space is entirely devoid 
of ordinary matter. This mode of transfer of heat is exemplified 
in the transfer of heat and light to the earth from the sun, and 
it is called heat radiation. The transfer of heat by radiation 

*This term is not here used in the narrow sense defined in Art. 43. 

398 



TRANSFER OF HEAT. 399 

shows itself by the burning sensation on one's face when one 
stands near an open fire when the air itself is cold. The transfer 
of heat by radiation is effected by wave motion like the wave 
motion which constitutes light, and these waves are transmitted 
by a medium, the ether, which fills all space. The molecular 
commotion in a hot body produces a commotion in the immedia- 
tely adjacent ether, this commotion spreads out in all directions 
as a wave disturbance, and when these waves impinge on a cold 
body, they produce molecular commotion in it, thus heating it. 
This wave commotion in the ether is called radiant heat.* 

Generally transfer of heat takes place by all three of the above 
processes simultaneously. Thus heat is distributed throughout 
a room from a hot stove partly by radiation (for one can feel 
with the hand the heat of the stove at a distance even though 
the air next the hand is quite cold) , by currents of air (convec- 
tion) , and by conduction. The last process, however, is almost 
unappreciable in air inasmuch as air is a very poor heat con- 
ductor. 

65. Temperature gradient. Fourier's law of heat conduction. 

The difference in temperature of a substance at two points di- 
vided by the distance apart of the points is called the temperature 
gradient between the points. Thus, one side of a wall is at o°C, 
the other side is at 75°C, the wall is 25 centimeters thick, and 
the temperature gradient through the wall is 3 degrees per centi- 
meter. Let t be the temperature of a substance at a point A, 
and let t -{- At be the temperature of the substance at an adjacent 
point B distant Ax from A. Then At /Ax is the temperature 
gradient between A and B. 

The conductive flow of heat in a substance is always associated 
with a temperature gradient, the temperature of the substance 
falls off in the direction of the flow of heat or, in other words, heat 
flows from the warmer to the colder parts of the substance. 
Consider a metal rod of sectional area q along which heat is 

*A fairly complete discussion of radiant heat is to be found in Franklin and 
MacNutt's Light and Sound, Appendix B, pages 301-316. 



400 THEORY OF HEAT. 

flowing by conduction. Let At /Ax be the temperature gradient 
along the rod and let H be the number of units of heat flowing 
past a given section of the rod in a given time r. Then H is 
proportional to q, to At/ Ax, and to the time r. We may, there- 
fore write 

E-Ka-g-r ( 32 ) 

in which K is a proportionality factor which is called the thermal 
conductivity of the given substance.* Thus, the thermal con- 
ductivity of copper at ordinary room temperature is about 0.95, 
that is to say, 0.95 calories of heat per second flow along a copper 
rod of which the sectional area is one square centimeter when the 
temperature gradient along the rod is one centigrade degree per 
centimeter. Expressed in terms of the same units the thermal 
conductivity of iron at ordinary room temperature is about 0.16; 
the thermal conductivity of glass at ordinary room temperature 
is about 0.0015 ; the thermal conductivity of parafnne at ordinary 
room temperature is about 0.0003 1 an d the thermal conductivity 
of flannel cloth expressed in terms of the same units is about 
0.000035. f The thermal conductivities of metals decrease with 
rise of temperature and the thermal conductivities of most other 
substances increase with rise of temperature. 

66. Flow of heat through a wall. Consider a wall or slab of a 
substance of thickness d, the faces of the wall being at tempera- 
tures /' and f respectively. Let q be the sectional area of the 
wall (at right angles to the direction of flow of heat through the 
wall) , and let K be the thermal conductivity of the substance of 
which the wall is made. Then (/' — f)/d is the temperature 
gradient through the wall, so that equation (32) gives 

H = K ^l-pi (33) 

*Methods for experimentally determining thermal conductivity are discussed 
in Preston's Theory of Heat, pages 505-572 (London, Macmillan and Company, 
1894). 

fSee Landolt and Bornstein's Physikalisch-Chemische Tabellen or Castell-Evans' 
Physico- Chemical Tables. 



TRANSFER OF HEAT. 



401 



in which H is the number of units of heat flowing through the 
wall in t seconds. 

67. Emission* of heat. Curve of cooling. A hot body gives 
off heat to the surrounding air and to surrounding bodies by 
convection, by radiation, and, to a very slight extent, by conduc- 
tion. The body is said to emit heat. The rate at which a body 
emits heat depends in a very complicated way upon the extent 



70 



60 



50 



2 
I 

5 40 



30 



















Dewar hattl* 






















































































































































































































































































































w 


mpc 


t 




































\ 


\ 



































































































































































































































10 20 30 40 50 60 70 80 90 I 

time in minutes 
Fig. 41. 

and character of the surface of the body, upon the nature of the 
surrounding gas and its freedom of motion, and upon the nature 
of surrounding bodies. For any given case a body may be heated 
and then as it cools its temperature may be observed at intervals, 

*This term as used in treatises on the theory of radiation refers to the emission 
of heat by a body by radiation only. Here the term means the giving off of heat 
by a body to the surrounding air and to surrounding bodies by conduction, by con- 
vection, and by radiation. 
27 



402 THEORY OF HEAT. 

and the result of such a series of observations may be plotted in 
the form of a curve which is called the cooling curve of the given 
body under the given conditions. Thus, Fig. 41 shows the 
cooling curve of an ordinary teapot filled at the start with hot 
water and allowed to stand on a table. The figure also shows 
the first part of the cooling curve of a Dewar bottle* containing 
hot water. 

Problems. 

69. A boiler shell is made of iron one centimeter thick, and 
20,000 calories per hour flow through each square centimeter of 
the shell. What is the temperature difference between inner and 
outer surfaces of shell? What would this temperature difference 
be with a copper shell of the same thickness? Use values of 
thermal conductivity given on page 400. Ans. 34°.7 C. ; 5 . 78 C. 

70. The inside surface of the window-glass in a house is at a 
temperature of I5°C, the outside surface is at a temperature of 
— io°C, and the glass is 4 millimeters thick. Calculate the heat 
in calories which during 24 hours flows out of the house by con- 
duction through a total window-glass area of 20 square meters 
and reduce the result to kilograms of coal at 6000 calories per 
gram. Use the value of thermal conductivity given on page 400. 
Ans. 1728 X io 6 calories; 288 kilograms of coal. 

71. The thermal conductivity of iron is 0.16 when heat flow 
is expressed in calories per second per square centimeter and tem- 
perature gradient in centigrade degrees per centimeter. What 
is the thermal conductivity of iron when heat flow is expressed 
in British thermal units per square inch per second and tempera- 
ture gradient in Fahrenheit degrees per inch? 

72. A metal vessel containing 25,200 grams of water has a flat 
face 30 X 30 centimeters which is pressed against the outside of 

*The Dewar bottle is a double-walled glass bottle with the air exhausted from 
the space between the walls. The inner surfaces of the walls are usually silvered. 
The Dewar bottle is sold under various trade names such as the "Thermos" bottle 
or the "Hotakold" bottle. The remarkable property of the Dewar bottle is ex- 
plained in Franklin and MacNutt's Light and Sound Appendix B. 



TRANSFER OF HEAT. 403 

a furnace wall made of brick, and the temperature of the water is 
observed to rise 7°-5C. in twenty minutes. The wall is 30 centi- 
meters thick, the inner face of the wall is at a temperature of 
I500°C. and the outer face of the wall is at 25°C. What is the 
thermal conductivity of the material of which the wall is made? 
Ans. 0.00356. 



INDEX. 



Absolute temperatures, definition of, 

284 
Acceleration, discussion of, 43 

displacement and velocity, 78 
Action and reaction, 5, 62 
Active forces and inactive forces, 120 
Adhesion and cohesion, 232 
Adiabatic expansion and compression, 

384 
^Eolotropy, 190 
JEiher and Matter, Larmor, 75 
Air thermometer, the, 282 
Ammonia refrigerating machine, the, 
322 
vapor, pressures and temperatures 
of, table of, 320 
Angular acceleration, 143 
momentum, 149 

conservation of, 149 
velocities, addition of, 176 
velocity, 143 
Annealing of steel, 338 
Archimedes' principle, 226 
Astronomy, Young's, 23 
Atmospheric moisture, 332 
Atomic theory of gases, 343 

of heat and thermodynamics, 
2 73 
Automobile engines, gyrostatic action 

of, 171 
Avogadro's principle, 340 

Balanced force actions, 50 

Balancing of rotating machine part, no 

Barometer, the, 224 

Baseball," curved flight of, 90, 253 

bat, the problem of, 163 
Beam, bent, discussion of, 196 
Beaume hydrometer, the, 232 
Belt, tension, 94 
Bent beam, discussion of, 196 
Bernoulli's principle, 251 

examples of, 252-256 
Boiler, tension in the shell of, 220 
Boiling point of a liquid, definition of, 

317 
points and melting points, 315 

table of, 329 
the phenomenon of, 320 
versus evaporation, 332 



Boltzmann's Vorlesungen iiber Gas Theo* 

rie, 346 
Bourdon gauge, the, 226 
Boyle's law, 206 

formulation of, 284 
Boynton's Kinetic Theory of Gases, 346 
British thermal unit, definition of, 308 
Brownian motion, the, 280 
Buckingham's Thermodynamics, 362 
Bulk modulus of a substance, 205 
Buoyant force of fluids, 226 

Calculus, differential and integral, meth- 
ods of, 38 
Calorie, definition of, 308 

the standard, 308 
Calorimeter, the water, 307 
Calorimetry, 304 
Capillary phenomena, 232 
Carnot C3 T cle, the, 377 
Carnot's cycle, application of, to the va- 
porization of water, 390 
to the freezing of water, 
392 
Castell-Evans' Physico-Chemical Tables, 

310 
Center of gravity, 61 

of mass, 61, 77 

of mass, equations of, ill 
motion of, 108 

of percussion, 163 
Centrifugal drier, 94 
C.g.s. system of units, the, 28 
Chemical compounds and mixtures, 339 
Circular mil, the, 20 

motion, 91 
Clausius' theorem, 380 
Clement and Desormes' determination 
of ratio of specific heats of a gas, 388 
Clock, the, 24 

Closed system, definition of, 105 
Coexistent phases, 338 
Cohesion and adhesion, 232 
Combining ratios, chemical, 339 
Combustion, heat of, 311 
Components of a force, 35 
Compound and elementary substances, 

339 
Compressibility, coefficient of, defini- 
tion, 206 



404 



INDEX. 



405 



Compressibility, coefficients of , table, 206 

of gases, 206 
Conduction, convection, radiation, 398 

of heat, Fourier's law of, 399 
Conductivity, thermal, definition of, 400 
Configuration, definition of, 105 
Connected system, the, 106 
Conservation of angular momentum, 149 

of energy, 132 

of momentum, 107 
Conservative systems, 130 
Constant and variable quantities, 36 

velocity, 143 
Constrained expansion of gases, 344 
Continuity, principle of, 37 
Convection, definition of, 297 

conduction, radiation, 398 
Cooling, curve of, 401 
Coriolis' law, 168 
Couple, definition of, 59 
Critical density, 330 

pressure, 330 

states, 330 

temperature, 330 
Cubic expansion, coefficient of, 291 
Curved flight of baseball, 90 
Curves of railways, discussion of, 95 
Cycle of vibrating body, 100 
Cyclic process, 377 
Cyclone, explanation of the, 296 
Cyclones and tornadoes, 246 
Cyclonic movements, 245 

D'Alembert's principle, 64 

Dalton's law, 331 

Degeneration, thermodynamic, 353, 358 

Degradation of energy, 278 

Degree, the, 19 f 

Densities of mercury, table of, 295 

of water, table of, 295 
Density and specific gravity, 23 
Derived and fundamental units, 27 

units, dimensions of, 29 
Dew-point, definition of, 332 
Differential calculus, method of, 38 
Dimensions of derived units, 29 
Discharge rate of a stream, 246 
Disk paradox, the, 252 
Displacement, velocity and acceleration, 

78 
Dissipation of energy, the, 278 
Dissociation pressures, 333 
Distectic points, definition of, 326 
Distillation, fractional, 339 
Distributed quantities, 40 
Drier, the centrifugal, 94 
Dynamic units of force, 81 

Easement curve, definition of, 97 



Eddy friction of fluids, 262 

Edser's Heat for Advanced Students, 279 

Efficiency, definition of, 125 

Efflux of liquid from a tank, 250 

Elastic fatigue, 204 

hysteresis, 203 

lag and viscosity, 203 

limit, 189 

modulus, see stretch modulus 
Elasticity, 183 
Electric Waves, Franklin, 45 
Elementary and compound substances. 

339 
Emission of heat, 401 
Endothermic reactions, 312 
Energy and work, 118 

conservation of, 132 

definition of, 126 

dissipation of, 278 

kinetic, 127, 128 

of a rotating body, 149 

potential, 127, 129 
Entropy of a gas, 389 

of a substance, 380 
Equilibrium, first condition of, 52 

second condition of, 56 

thermal, 273 
Erg, definition of, 122 
Eutectic alloys, 326 

mixtures, definition of, 325 

point, the, 325 
Evaporation versus boiling, 332 
Exothermic reactions, 312 
Expansion, cubic, coefficient of, 291 

free and constrained, of gases, 344 

linear, coefficient of, 289 

of gases, thermal, 281 

of solids, peculiarities of, 292 

of water and mercury, 293 

thermal, of liquids and solids, 287 

Falling bodies, 86 

Fatigue, elastic, 204 

First law of thermodynamics, 278, 304 

Flow, lamellar, 241 

permanent and varying states of, 
240 

simple, 240 
Fluid friction, 261 

practical formula for, 263 

the ideal frictionless incompressible, 
248 
Fluids and solids, 188 
Foot-pound, definition of, 122 
Force and its effects, 75 

components, 35 

dynamic units of, 81 

gravitational units of, 82 

measurement of, 84 



406 



INDEX. 



Force, moment of, 51 

parallelogram, the, 32 

polygon, the, 33 

resolution of, 34 

types of, 76 
Forces, active and inactive, 120 

parallelogram of, proof of, 134 
Franklin, Crawford and MacNutt's 

Practical Physics, 287 
Franklin's Electric Waves, 45 
Free expansion of gases, 344 
Freezing mixtures, 327 
Frequency of harmonic motion, 100 
Friction, angle of, 119 

coefficient of, 119 

eddy, of fluids, 262 

fluid, 118, 261 

rolling, 118 

sliding, 118 

viscous, of fluids, 262 

work and energy, 118 
Fourier's law of heat conduction, 399 
Fundamental and derived units, 27 
Fusion, latent heat of, 328 

Gas, definition of, 189 
specific heats of, 383 

Gases, atomic theory of, 343 
compressibility of, 206 
mixed, pressures of, 331 
thermal expansion of, 281 

Gauge tester, the, 226 

Gauging of streams, 260 

Gay Lussac's law, 281 

formulation of, 284 

Gee-pound, the, definition of, 84 

Geometric addition, see vector addi- 
tion 

Gradient, definition of, 41 

Gravitational units of force, 82 

Gravity, center of, 61 

determination of, 158 

table of accelerations of, 160 

pendulum, the, 156 

Gyration, radius of, 148 

Gyroscope, the, 164 

Hardening of steel, 337 
Harmonic motion, 99 
rotatory, 155 
Heat, atomic theory of, 273 

emission of, 401 

measurement of, 306 

mechanical equivalent of, 308 

of combustion, 311 

transfer of, 398 

A Mode of Motion, Tyndall, 274 

capacity, specific, 310 

conduction, Fourier's law of, 399 



Heat, conductivity, definition of, 400 

for Advanced Students, Edser, 279 
Heating of water by dissipation of 

energy, 304 
High Temperature Measurement, Le 

Chatelier & Boudouard, 285 
Hodograph, definition of, 44 

of a projectile, 88 
Holman's Precision of Measurements, 13 
Hook's law, 189 
Hoop, the rotating, 97 
Horse-power, definition of, 123 
Horse-power-hour, definition of, 125 
Humidity of the air, 333 
Hygrometry, 332 
Hydraulic paradox, the, 252 

ram, the, 243 
Hydraulics, 239 
Hydraulics, Merriman, 260 
Hydrometer, the, 229, 231 
Hydrostatic press, the, 220 

pressure, 204 
Hydrostatics, 219 
Hysteresis, elastic, 203 

Impact, discussion of, 107 

Impulse, definition of, 161 

Inertia, definition of, 81 

Injector, the steam, 253 

Instantaneous axis, definition of, 174 

Integral calculus, method of, 38 

Invar alloy, 292 

Irreversible processes or sweeps, 355 

Isentropic expansion and compression, 

384 
Isothermic expansion and compression, 

384 
Isotropic strain, 204 
Isotropy, 190 

Jet, see water jet 

pump, the, 252 
Johnson's Materials of Construction, 200 
Joule, definition of the, 122 
Joule and Thompson's principle, 343 

Kater's pendulum, 158 
Kilogram, the, 21 

of force, the, 22 
Kilowatt-hour, definition of, 125 
Kinematics of a rigid body, 174 
Kinetic energy, 127, 128 

of a rotating body, 149 
theory of gases, see atomic theory 
of gases 

Lag, elastic, 203 

Landolt & Bornstein's Physikalisch- 

Chemische Tabellen, 286 
Larmor's JEther and Matter, 75 



INDEX. 



407 



Latent heat, definition of, 328 

heats, table of, 329 
Laws of motion, 79 
Least Squares, Merriman, 13 
LeChatelier & Boudouard's High Tem- 
perature Measurement, 285 
Linde's liquid air machine, 348 
Linear expansion, coefficient of, 289 
Liquid air machine, Linde's, 348 

definition of, 189 

efflux of, from a tank, 250 

energy of, 248 
Lockyer's Stargazing, 24 
Locomotive, behavior of, on curves, 95 

nosing action of, 97 
Longitudinal strain, potential energy of, 

195 
Low barometer, definition of, 296 

Mach's Mechanics, 75 
Magdeburg Hemispheres, the, 219 
Manometer, open-tube, 225 
Manometers, 225 
Mass, center of, 61, 77 

equations of, in 
motion of, 108 

definition of, 20 
Material particle, 76 
Materials of Construction, J. B. Johnson, 

200 
Maximum pressure of a vapor, 317 
Measurement, general discussion of, n 

of force, 84 
Measures and units, 26 
Mechanical equivalent of heat, 308 
Mechanics, Mach, 75 

Theoretical, Ziwet, 50, 75 
Melting point of a substance, definition 
of, 316 

points and boiling points, 315 
table of, 329 
Mercury-in-glass temperatures, 285 
Mercury-in-glass thermometer, 284 
Merriman's Hydraulics, 260 

Least Squares, 13 
Metacenter, definition of, 230 
.Meter, the, 19 

Minchin's Treatise on Statics, 50 
Minimum temperature of a vapor, 317 
Mixed gases, pressures of, 331 
Mixtures and chemical compounds, 339 
Modulus, stretch, of a substance, 193 
Moisture, atmospheric, 332 
Moment of force, definition of, 51 

of inertia, definition of, 144 
Moments of inertia about parallel axes, 
150 
comparison of, 155 
table of, 148 



Momentum, conservation of, 107 

definition of, 106 
Motion in a circle, 79, 91 
laws of, 2 
rotatory, 143 
translatory, 143 

and rotatory, correspondence 
between, 154 
types of, 76 

Nernst's Theoretical Chemistry, 330 
Newton's laws of motion, 2, 79 

Orbit, definition of, 43 
of a projectile, 88 

Parallelogram of forces, proof of, 134 

the, 32 
Particle, material, 76 
Particles, systems of, 105 
Pascal's principle, 220 
Pendulum, the gravity, 156 

the reversion, 158 

the simple, 103, 157 

the torsion, 155 
Percussion, center of, 163 
Perfect gas, definition of the, 349 
Perigon, the, 19 

Perpetual motion impossible, 131 
Phases, coexistent, 338 
Physical science, subdivision of, 350 
Phy si co-Chemical Tables, John Castwell- 

Evans, 310 
Physics, the science of, 13 
Physikalisch Chemische Tabellen, Landolt 

& Bornstein, 286 
Pitot tube, the, 259 
Planck's Vorlesungen iiber theoretische 

Physik, 346 
Poisson's ratio, 193 
Polygon of forces, 33 
Potential, definition of, 45 

energy, 127, 129 
Pound, the, 21 

of force, the, 22 
Power, definition of, 123 

transmission of, 123 
Practical Physics, Franklin, Crawford 

and MacNutt, 287 
Precession, definition of, 164 

examples of, 169 
Precessional rotatory motion, 164 
Precision of Measurements, Holman, 13 
Pressure, measurement of, 225 

at a point in a fluid, 219 

gauges, 225 

in a liquid due to gravity, 221 
Principle of D'Alembert, 64 

of continuity, 37 



4o8 



INDEX. 



Principle of the conservation of momen- 
tum, 107 

of virtual work, 64, 133 
Projectiles, 87 
Process, cyclic, 377 

curve, 376 
Pure torque, 59 

Radian, the, 20 

Radiation, conduction, convection, 398 

Radius of gyration, 148 

Railway curves, discussion of, 95 

Range of a projectile, 88 

Rate of change, definition of, 37 

Recalescence of steel, 292, 336 

Refrigerating machine, the ammonia, 
322, 366 

Regelation, 322 

Resilience, definition of, 202 

Resolution of vectors, 34 

Resultant of a number of forces, 60 

Retarded transformations, 337 

Reversible processes, 354 

Reversion pendulum, the, 158 

Rigid body, 76 

kinematics of, 174 

Rod, twisted, discussion of, 211 

Rolling friction, see friction 

of ships, prevention of, 172 
wheel, equivalent mass of, 151 

Rotating hoop, the, 97 

Rotatory harmonic, 143 
motion, harmonic, 155 
precessional, 164 
and translatory motion, cor- 
respondence between, 154 

Rowland's work on the heating of water, 

305 
Rubber, thermo-elastic properties of, 

393 

Second, definition of the, 23 

law of thermodynamics, 350, 363 
Scalar and vector products and quo- 
tients, 35 
quantities, 30 
Schlick's device for the prevention of 

rolling of ships, 172 
Science of Physics, the, 13 
Sensitive flame, the, 243 
Shearing stress, 207 
Simple pendulum, the, 103, 157 
Slide modulus of a substance, 210 
Slug, the, definition of, 84 
Solids and fluids, 188 
Specific gravity and density, 23 

heat, definition of, 310 

heats of a gas, 383 
Spin acceleration, 143 



Spin momentum, 149 

conservation of, 149 
velocities, addition of, 176 
Statics, 50 

Treatise on, Minchin, 50 
Stargazing, Lockyer, 24 
Steam engine, the, 366 
total heat of, 329 
Steel, annealing of, 338 
hardening of, 337 
physical properties of, table, 202 
recalescence of, 292 
tempering of, 338 
Strain, isotropic, 204 

potential energy of, 206 
longitudinal, 192 
principal stretches of, 213 
Stream, discharge rate of, 246 

lines, of a fluid, 240 
Streams, gauging of, 260 
Stress, principal pulls of, 213 
and strain, 183 

general equations of, 213 
longitudinal, 191 
practical relations of, 200 
shearing, 207 
types of, 191 
Stresses and strains, homogeneous and 
I non-homogeneous, 1 83 
Stretch modulus, determination of, 194 
table of values of, 194 
of a substance, 193 
Sublimation, 330 

Superheating and under-cooling of liq- 
K uids, 330 
Surface tension, measurement of, 234 

of liquids, 233 
Sweeping processes, 355 
System, closed, definition of, 105 
connected, the, 106 
of units, the c.g.s., 28 
Systems, conservative, 130 
of particles, 105 

Table, coefficients of compressibility, 

206 
densities of mercury, 295 

of water, 295 
melting points and boiling points, 

latent heats, 329 
moments of inertia, 148 
physical properties of steel, 202 
pressures and temperatures of 

water vapor, 320 
of accelerations of gravity, 160 
values of stretch modulus, 194 
vapor pressures and temperatures 

of ammonia, 320 
Temperature, measurement of, 282 



INDEX. 



409 



Temperature ratio, definition of, 360 

and thermal equilibrium, 273 
Temperatures, absolute, definition of, 

284 
Tempering of steel, 338 
Tensile strength, definition of, 201 
Tension in the shell of a boiler, 220 

surface, of liquids, 232 
Theoretical Chemistry, Nernst, 330 
Thermal conductivity, definition of, 400 

equilibrium, 273 

expansion of liquids and solids, 287 

expansion of gases, 281 
Thermodynamic degeneration, 353, 358 
Thermodynamics and atomic theory of 

heat, 273 
Thermodynamics, Buckingham, 362 
Thermodynamics, first law of, 278, 304 

second law of, 350, 363 
Thermo-elastic properties of rubber, 393 
Thermometer, the air, 282 

the mercury-in-glass, 284 
Top, the, 166 

Tornadoes and cyclones, 246 
Torque, definition of, 51, 134 

pure, 59 
Torques, addition of, 177 
Torricelli's theorem, 251 
Torsion pendulum, the, 155 

of a rod, 211 
Total heat of steam, 329 
Transfer of heat, 398 
Transformations, retarded, 337 
Transition temperatures, 334 

definition of, 292 
Translatory motion and rotatory mo- 
tion, correspondence between, 154 
Twisted rod, discussion of, 211 
Tyndall's Heat a Mode of Motion, 274 
Types of force, 76 

of motion, 76 

Under-Cooling and superheating of liq- 
uids, 330 
Units and measures, 26 

fundamental and derived, 27 

van der Waals' equation, 348 
Vapor pressure, 332 

pressures, maximum, 317 



Vapor pressures and temperatures of 
ammonia, table of, 320 

and temperatures of water, table, 
of, 320 

temperatures, minimum, 317 
Vaporization, latent heat of, 328 
Variable and constant quantities, 36 
Variations in space, 40 

in time, 37 
Vector addition, 31 

and scalar products and quotients, 
35 
quantities, 30 

space variation of, 45 
Vectors, resolution of, 34 

variations of, 42 
Velocity, discussion of, 43 

displacement and acceleration, 78 
Venturi water meter, the, 257 
Virtual work, principle of, 64, 133 
Viscosity, elastic, 203 

of a fluid, definition of, 265 
Viscous friction of fluids, 262 
Vortex rings, 244 

Water calorimeter, the, 307 

hammer, the, 242 

heating of, by dissipation of energy, 
304 

jet, impact of, 257 
reaction of, 257 

vapor, pressures and temperatures, 
table of, 320 
Watt, definition of, 123 
Watt-hour, definition of, 125 
Watt's diagram, 376 
Weights and measures, 19 
Wheel, rolling, equivalent mass of, 151 
Work and energy, 118 

definition of, 120 

power- time units of, 125 

units of, 122 

virtual, principle of, 64, 133 

Yard, the, 19 

Yield point of steel, 201 

Young's modulus, see stretch modulus 

Young's General Astronomy, 23 

Ziwet's Theoretical Mechanics, 50, 75 



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